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Second Edition, in Two Volumes 




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SINCE the present Translation of Professor Lotze's 
* System der Philosophic ' was begun, both the author him- 
self, w^o cordially welcomed the undertaking, and Pro- 
fessor Green, who first definitely proposed it, have been 
removed by death. These two distinguished men, however 
different in method and style of thought, had some funda- 
mental tendencies in common ; and it may be of interest to 
Professor Lotze's admirers in this country to know that 
Professor Green not only executed an important part of the 
Translation 1 , but intended to take upon himself the task of 
revising and editing the whole 2 , which was not entrusted to 
the present Editor till after Professor Green's death. 

The Translation of the Logic has been throughout 
adapted to the second edition. But the Author's intended 
revision of the Metaphysic was not carried out, and the 
projected Part III of the * System der Philosophic 7 was 
never written. What the Author made known of his 
intentions in these respects is mentioned in the Prefaces to 
the Metaphysic. 

The translation of Part I, the Logic, has been executed 
by several hands; the whole of Book I by Mr. R. L. 
Nettleship, Fellow of Balliol College, Oxford; Book II, 

1 See Preface to the * Metaphysic.' 

2 He said to the present Editor : * The time which one spent on such 
a book as that (the " Metaphysic") would not be Wasted as regards 
one's own work.' 


Chapters i-v (inclusive), by Mr. F. H. Peters, Fellow of 
University College, Oxford (with the exception of the ' Note 
on the j^ogical Calculus/ which was translated by the 
Editor) ; chapters vi-ix (inclusive), by Mr. F. C. Conybeare, 
Fellow of University College ; and chapter x by the Editor ; 
and the whole of Book III by Mr. R. G. Tatton, Fellow of 
Balliol College. 

The Editor has revised the whole translation, and is 
responsible in all cases for the rendering finally adopted. 
He has to thank Mr. J. C. Wilson, Fellow of Oriel College, 
Oxford, for the most cordial and ample assistance in deal- 
ing with the numerous passages in which mathematical 
knowledge was required. It is believed that the translation 
of these passages will, owing mainly to his help, be found 
on the whole correct and intelligible. 

The Table of Contents was furnished by the several 
translators for their respective portions. It should be 
observed that the original Table of Contents supplies a few 
headings (in Book I only), besides those of the chapters. 
These are distinguished from the headings supplied by the 
translator by being printed in italics. The Index was added 
by the Editor. 

No endeavour has been made to introduce uniformity of 
style into the different portions of the translation. But in 
the case of a few important technical terms it has been 
thought advisable to introduce renderings as nearly uniform 
as the context would allow. Unavoidable variations in the 
translation of a German word, or ambiguities in the 
employment of an English one, are pointed out, to some 
extent, in the Notes and Index ; and in all cases references 
are freely give^ to any passages that explain the precise 
point of the Author's choice of words. It is hoped that by 


this means the reader may be assisted to master the some 
what subtle distinctions which govern the Author's usage, 
without the aid of a Glossary, which could indicate them 
but roughly. Still Professor Wallace's observations on the 
meaning of some German terms, prefixed to his translation 
of Hegel's shorter Logic, will be found useful by many 

Jn the case of two of the sections which treat of mathe- 
matical questions (234 and 237) the Editor found himself 
in a perplexity which could have been removed if the 
Author had been still living. The reasoning of sect. 234 
seemed more than doubtful ; while the Author himself had 
requested the suppression of sect. 237 as 'wholly errone- 
ous,' regretting that he had put forward such ( nonsense,' 
and explaining that he had been ' misled by the error of a 

This unqualified condemnation seemed on consideration 
hardly to apply to sect. 237, and to be such as might have 
been intended for sect. 234 ; but as the Author mentioned 
not only the number of sect. 237, but the pages on which 
it stands, the hypothesis of a mere clerical error is almost 
excluded. It is nevertheless conceivable that there may 
have been some misapprehension ; and therefore it has been 
thought advisable not to withdraw sect. 237 entirely, but to 
print it as an Appendix. 

In preparing the second English edition of the 'System 
der Philosophic ' no important alterations have been made 
in the translation, although a few verbal corrections have 
been found necessary. 


THOUGH I venture to describe the present work as the 
first part of a System of Philosophy, I hope that this desig- 
nation will not be supposed to indicate the same pretensions 
which it was wont to herald in times gone by. It is obvious 
that I can propose to myself nothing more than to set forth 
the entirety of my personal convictions in a systematic 
form ; such a form as will enable the reader to judge not 
only to what degree they are consistent with themselves, 
but also how far they are capable of serving to unite the 
isolated provinces of our certain knowledge, in spite of the 
great gaps that lie between them, into a coherent view of 
the world bearing the character of completeness. In the 
present volume, which begins my exposition, I have been 
guided, as I shall be in the others, by this purpose. 

In the First Book, although entirely rewritten, I have 
followed in essentials the line of thought of my short work 
on Logic of 1843, which has long been out of print. I 
have not seen reason to depart from this line, to which my 
own interest in the exposition of Logic is as much confined 
now as it was then. Now, as then, I consider it useless 
labour to attempt extensions and improvements of the 
formal part of Logic, within the limits of the general 
character which in fact and' of necessity attaches to it ; but 
whatever in it appeared worth knowing, if only as belong- 
ing in a certain sense to the history of culture, I have to 
the best of my belief conveyed without omission, and have 
taken pains to do so as simply as possible. 

The Second Book needs no preface ; it is quite free 


ftopci the bonds of system, and simply puts together what- 
ever I thought useful. The selection of matter might be 
different in many parts, a great deal might be adcted, and 
a great deal, it will be thought, might be spared. The 
reader should regard it as an open market, where he may 
simply pass by the goods he does not want. 

The original purpose of the Third Book has not been 
carried out. It was meant to treat of the subjects with 
which it does in fact deal, on the method of a historico- 
critical exposition of systematic logical views the .views 
which have appeared both in Germany and in several other 
countries in a variety of forms that demand a high degree 
of interest and appreciation. But it became clear on 
making the attempt that such a task could not be achieved 
within the limits of the present treatise, not, that is to 
say, with the thoroughness due to the valuable works in 
question. Another opportunity may possibly be found for 
it ; but in the meantime I was induced by the failure of 
this plan to dispense for the present with all reference to doc- 
trines which are not my own, and to put forward nothing 
but what either is common property, or belongs to my own 
individual mode of viewing the subject. I trust that the 
whole of my doctrine is not merely of this latter kind ! 
GOTTINGEN : Jttnc JO, 1874. 

The present (2nd) edition contains a number of improve- 
ments in detail, and a single addition of some length, the 
c Note on the Logical Calculus/ p. 208 (E. Tr. vol. i. p. 275). 
I may remark with reference to p. 222 (E. Tr. vol. i. p. 297) 
that Jevons speaks of Potassium. Perhaps the reader can 
conjecture why J have preferred to speak of Sodium. 

GOTTINGEN: September 6, 1880. 



Of Thought (Pure Logic). 


Section I. Coherent and coincident ideas ..... i 

,, If. Necessary connexion of ideas generally ... 2 

,, III. Connexion of ideas established by thought ... 3 

,, IV. Is thought an activity rather than an event? . . 3 
V. Or, if an activity, has it any specific function in giving 

truth? 4 

,, VI. Thought in man compared with the other animals . 4 
,, VII. Thought adds the notion of coherence to connected 

ideas ......... 6 

,, VIII. Have the forms of thought a real validity? ... 7 
IX. Provibional answer, assuming thought to be a means 

to knowledge ........ '8 

,, X. Logical distinguished fiom Psychological enquiries . 10 
,, XI. Order of enquiry. Pure or formal logic . . .10 

XII. Applied logic 1 1 

XIII. Knowledge . . . . . . . .12 



A. The formation of impressions into ideas. 

1. Conversion of impressions into ideas the first work of thought . 13 

2. This is effected by the logical act of naming .... 14 

3. Which may be called an act of objectification . . 14 

4. In the specific forms of the parts of speech . . . .16 

5. Relation of substantive, verb, adjective, to substance, event, 

property . . . . . . . . 17 

6. Relation of thought to its linguistic expression . . 19 

7. The other parts of speech. Prepositions and conjunctions . 20 

8. Judgment not rightly treated in pure logic before conception . 2 a 



B. Position, distinction, and comparison of the matter of simple ideas. 

9. Further r reaction of thought in positing, distinguishing, and 

comparing ideas ........ 24 

10. Position ' ; practically inseparable from ' objectifi cation * 

.. .. 

11. Position implies and renders possible distinction ... 26 

12. Comparison ; supposed to involve falsification of impressions 

by generalisation . . . . . . . 2 7 

13. Such a view ignores the true logical ixupoit of the act . . 28 

14. Comparison implies a 'universal'; this not a universal 'con- 

cept' .......... 30 

15. Like all universals, it is not strictly an ' idea ' . . 31 

16. Distinction of instances of a universal implies idea of more 

and less .......... 32 

17. Also those of unity and multiplicity, and greatness and small- 

ness .......... 33 

18. Logic is not concerned with the origin of these ideas, or with 

deductions from them ....... 34 

19. In the operations of (B) as compared with those of (A) thought 

may be called receptive ...... 35 

C. The formation of the concept. 

20. Synthesis in consciousness : its various forms . . -37 

21. The form of ' conception ' implies the idea of a ground 'for the 

synthesis ......... 38 

22. Comparison of different instances and observation of the same 

instance under different circumstances .... 40 

23. * Abstraction ' involved in comparison does not mean mere 

omission of differences ....... 41 

24. The formation of the second universal (logical concept) implies 

the first (14) ........ 42 

25. Terminology : ' content/ ' extent,' ' co-ordination,' * subordina- 

tion,' * subsumption ' ...... 44 

26. Concept ' is rightly applied to individual things ... 45 

27. Universals not necessarily concepts; they may be general 

images .......... 46 

28. Marks not merely co-ordinated, but also mutually determined, 

in a concept ......... 47 

29. Subordination of species to genus, and subsumption of species 

and genus under a mark ....... 49 

30. Species = a universal which can be imaged, genus = one 

which can only be formulated ..... 50 



31. Inverse ratio of content and extent, how far true or important "* 51 

32. Kinds of marks : ' differentia,' ' property,' ' accident * . . 53 

33. Idea of a complete system of concepts . . v -55 

Transition to the Form of the Judgment. 

34. Change makes such an idea unrealisable, and leads from con- 

ception to judgment 56 

35. Conception itself involves questions which also lead to judg- 

ment 57 



Preliminary observations on the meaning and customary division 
of judgments. 

36. Judgment expresses a relation between the matter of two ideas 59 

37. This is indicated in the terms Subject, Predicate, and Copula 60 

38. Judgments differ according to the different senses of the copula 61 

39. The logical sense of the copula is not affected by the quantity 

of the judgment 62 

40. Nor by its quality ......... 63 

41. Nor by its modality ', as ordinarily understood ... 64 

42. True apodeictic modality is found in the three forms of relation 66 

43. Not of course that any form of judgment can guarantee its 

material truth 67 

44. So-called problematic judgments are not truly problematic, 

nor are questions or prayers ...... 68 

45. The only true problematic modality is expressed by particular 

and singular judgments ....... 70 

46. The ordinary classification omits or confuses many modal 

distinctions ......... 70 


A. The imperson il judgment ; the categorical judgment\; 
the principle of identity. 

47. The categorical judgment is logically preceded by the im- 

personal ......... 72 

48. Which does not express mere perception, but implies logical 

activity .......... 73 

49. Relating a present perception to a permanent though un- 

expressed subject ' . -74 

50. Difficulty as to the logical import of the categorical judgment 75 




51. It does not mean the identity of the subject and predicate . 75 

52. Nor is it explained by saying that the one is predicated of t the 

othr 76 

53. Nor by reference to the metaphysical relation of substance 

and attribute ......... 78 

54. In fact, the judgment is indefensible against the principle of 

identity .......... 79 

55. The logical interpretation of that principle .... 80 

B. The particular judgment ; the hypothetical judgment ; 
the principle of sufficient reason. 

56. The difficulty applies to analytical as well as to synthetical 

judgments . . . . . . . . .81 

57. The justification of categorical judgments is that they are 

really identical . . . . . . . 83 

58. Illustrations 83 

69. But in that case they are T\Q\. judgments at all in the real sense 86 

60. This dilemma is met in the hypothetical judgment by making 

the identity conditional . . . . . . .88 

61. The idea of a condition implies the assumption of a general 

coherence of things ....... 89 

62. Logical possibility and meaning of such coherence : cause and 

reason .......... 90 

63. True formulation of the ' principle of sufficient reason ' . . 91 

64. The principle of identity alone is no source of knowledge . 93 

65. The princ. rat. stiff, not a necessity of thought, but a fact of 

all mental experience ....... 94 

66. Responsiveness of thinkable matter to thought illustrated from 

19 9 6 

C. The general judgment ; the disjunctive judgment ; the dictum de 
omni et nullo and the principium exclusi medii. 

67. The connexion between reason and consequence must be uni- 

versal .......... 96 

68. This universality is expressed in the ' general ' judgment . 97 

69. Further determination of the predicate in the disjunctive 

judgment 99 

70. True formulation of dictum de omni et nullo . , . .100 

71. Princ. excl. med. is only one case of the ' law of disjunctive 

thought* 101 

72. Its true logical formulation . . . . . . .102 


PAG** 1 

73. Incompatibility of contrary, and compatibility of disparate, ^ 

predicates . . . . . . . . 103 

74. The disjunctive judgment leads on to inference . j .105 

Appendix on immediate inferences. 

75. Inference ad subalternatam . . . . . . .106 

76. Ad subalternantem . . . . , . . .107 

77. Ad contradictor iam . . . . . . . .108 

78. Ad contrariam and ad subcontrariam . . . . .108 

79. Inference by conversion . . . . . . .109 

80. Conversion of universal judgments . . . . .110 

81. Conversion of particular judgments . . . . . in 

82. Conversion by contraposition 112 


Preliminary remarks upon the Aristotelian doctrine of Syllogism. 

83. Formation of the four figures 114 

84. General conditions of valid inference in them . . .115 

85. Special conditions in each figure. The first figure . .116 

86. The second figure . . . . . . . 117 

87. The third figure, when both premises are affirmative . .118 

88. The third figure, when the premises are mixed . . .118 

89. The third figure, when both premises are negative . .119 

90. The fourth figure is superfluous 120 

91. Superiority of the first to the other figures . . . .121 

92. Reduction of the other figures to the fiist . . . .122 

93. Syllogisms with hypothetical premises involve no new 

principle . . . . . . . . .123 

94. Difference of the relation between reason and consequence from 

that between cause and effect . . . . . -125 

95. Syllogisms with disjunctive, copulative, or remotive premises 1 26 

96. Chains of inference 127 

A. Syllogistic inferences ; inference by subsumption ; inference by 
induction ; inference by analogy. 

97. The Aristotelian or subsumptive syllogisms merely make 

explicit what is implied in the disjunctive judgment . 128 

98. Such inference by subsumption involves a double circle . 129 



9 9. Illustrations when the premises are (i) analytical, (2) syn- 
thetical 13 

100. It must be possible to establish (i) major and (2) minor 

premises without full knowledge *3 2 

101. Inductive inference as solution of the first requirement . .133 

102. The defect of induction (as of subsumption) lies in the 

practice, not the principle, of it 135 

103. Inference by analogy as solution of the second requirement . 137 

104. Defect and justification of analogical inference . . .138 

B. Mathematical inferences ; inference by substitution ; inference 
by proportion ; constitutive equation. 

105. The previous forms of inference deal only with universal 

subjects and predicates . . . . . . .140 

106. Thus do not satisfy the needs of real thinking, which requires 

them to be specific 141 

107. They are in fact inferences from the extent, instead of from 

the content, of concepts . . . . . . .142 

108. Inference from content, though implying experience, yields 

results for logic . . . . . . . .143 

109. It implies substitution of an analysed for an unanalysed 

middle term 143 

110. Remarks on the symbolisation of logical relationships . . 145 

111. Inference by substitution is only strictly applicable to pure 

quantities . . . . . . . . .146 

112. Still, as an ideal of thought in general, it has its place in 

logic 147 

118. Extension of it to incommensurable objects in the form of 

proportion 148 

114. Illustration from Geometry 150 

115. Limitation of inference by proportion. Ultimate disparity of 

things 151 

116. Proportion between marks is modified by the constitution of 

the whole subject 153 

117. Inference by proportion thus leads to the idea of constitutive 

concepts . .154 

118. Which however are only fruitful in Mathematics, where all 

is commensurable 155 

119. To deal with disparate marks, we must go on to classifi- 

cation. 157 


C. The systematic forms : classification ; explanatory theory ; , 
the dialectic ideal of thought. 


120. ' Concept '= not a mere sum of marks, but a sum connected 

according to a rule .157 

121. Many such concepts are formed unconsciously by 'psychical 

mechanism' ......... 158 

122. Hence the idea that any group of common inarms forms a 

concept . . I59 

123. Whereas the true concept is found only in the union of essen- 

tial marks ......... 160 

124. The distinction of essential from unessential leads to com- 

parison and classification . . . . . .161 

125. Artificial or ' combmatory ' classification .... 162 

126. Its defects. It may include more than the facts . . .164 

127. Or less than the facts 164 

128. And it takes no account of the different values of marks m 

the concept 165 

129. Logical classification aims at ' constitutive ' concepts, or 

'ideas' . . . . . . . . . .167 

130. Which naturally connect with the notions of active tendency 

and purpose ......... 168 

131. And so with that of more and less perfect species . . 1 70 

132. Illustrations from Mathematics of the gradation of species . 172 

133. The logically most perfect species is that in which all the 

marks are at the highest value allowed by the genus . 1 74 

134. But each genus may itself have its standard of perfection in a 

higher genus . . . . . . . . .175 

135. And so lead ultimately to a highest genus which governs the 

development of all the rest . . . . . . 177 

136. Thus we get the ideal form of natural classification . .178 

137. Stationary and progressive perfection of species. ' Type ' 

and 'Ideal' . . . . . . . . . 179 

138. The form of classification by development (like other logical 

forms) is only an ideal . . . . . . . 1 80 

139. The development of concepts is conditioned by something 

other than the concepts themselves . . . . .181 

140. This condition must find a place in a complete logical 

system . . . . . . . . . .182 

141. No theory of ' emanation ' of one concept from another can 

dispense with it 183 

142. And indeed classification itself leads beyond the particular 

concept to universal laws of connexion between its marks 185 



K3. This does not contradict, but confirms, the previous repre- 
sentation of the concept as determining the connexion ot 
it% marks . . . . . . . . .186 

144. A thing is the result, according to universal laws, of the sum 

of its conditions ........ 187 

145. This view dominates modern science, which explains, in con- 

trast with ancient, which classifies. Mechanical character 

of the former . . . , . . . . .188 

146. Unsatisfactoriness of its ideal 190 

147. Antagonism between scsthetic and scientific view of things. 

Possible reconciliation . . . . . . .191 

148. 'Laws' are not external to reality, but constitute its very 

nature .......... 193 

149. Form of the ultimate ideal of thought . . . . 194 

150. Supposed analogy of the living organism. Hegelianism. 

1 Speculation ' ........ 195 

151. Value of the ' speculative * ideal. It belongs to logic, but 

points beyond it . . . . . . 197 


Applied Logic. 
152-3. Prefatory Remarks . . . . . . . . 199 



154. Ideas, how communicable ....... 20,2 

155. Poetry and rhetoric 203 

156. Uncertainty of communication ...... 204 

157- Explanation by abstraction ...... 205 

158. This the only method for simple ideas . . . .206 

159. Explanation by construction. Description .... 207 
160-1. Description and definition ...... 209 

162. Nominal and real definitions . . . . . -213 

163. Three faults to avoid 214 

164. Elegance and brevity . . . . . . . .216 

165. Evil of superfluity 218 

166. Popular definitions . . . . . . . .219 

167. Genetic definition . . . . . . . .220 

168. The end of definition is the conception , . .222 





169-70, We must start from the conceptions already expressed in 

language 225 

171. Disparate groups of sensations 227 

372. Popular language justified ....... 229 

173. Relations between the members of these groups. Tastes. 

Colours . . . . . . . . . -231 

174. Scale of sounds . . . . . . . . -233 

175. Heat-sensations . . . . . . . . -235 

176. Arbitrariness of scale ........ 236 

177. Illustrations from practical life . . . . . -237 

178. Moral and aesthetic distinctions ...... 239 

179. Transition from concave to convex ..... 241 

180. The distinctions remain in spite of the transition from one 

conception to another . . . . . . .242 

181. And though there be a term in the series that satisfies both 

conceptions ......... 243 

182. Illustrations ......... 245 

183. Development ......... 246 



^84. The notion of a universal scheme or system of conceptions . 249 

185. Pythagorean ism . . . . . . . . .250 

186. Grandeur of its general idea ...... 252 

187. Poverty of the particular form in which it is expressed . 254 
388. Numbers and things 255 

189. Other kindred speculations 257 

190. Demand for symmetry . . . . . . .259 

191-5. The Hegelian dialectic 262 

196. The scheme of Leibnitz 271 

197. Is such a scheme possible ? 273 

198. What it would require 275 

Note on the Logical Calculus . . . . . a 77 

b 2 





199. Discovery and proof. Proof of particular and of universal 

propositions . . . . . . . . . 299 

200. Proof rests on axioms. Axioms how distinguished . . 30 r 

201. Before starting to prove a proposition we must know that it 

is worth proving, i. e. that (a] the ideas are definite . 303 

202. () their combination possible ...... 304 

203. (Y) the proposition true in fact 306 

204. Eight forms of proof distinguished 307 

205-6. (i) First direct progressive proof from the conditions of*?"" 

to r 308 

207. (2) Second direct progressive proof from T to its conse- 

quences 312 

208. (3) First direct regressive proof from Tto its conditions . 313 
209-10. (4) Second direct regressive proof from the consequences 

of Tto T 315 

211. (5) First indirect progressive proof . . . . . 317 

212. (6) Second indirect progressive proof . . . . . 319 

213. Indirect regressive proofs 321 

214-5. What is ordinarily called proof by analogy is really proof 

by subsumption . . . . . . . -322 

216. The mathematician's proof by strict analogy is also proof by 

subsumption . . . . . . . . . 328 

217. Analogy and the Dictum de omni et nullo .... 330 



218. No rules for the discovery of a proof, but the problem itself 

may give a clue 333 

219. Illustiations from Geometry ...... 334 

220-5. The conditions of equilibrium ...... 336 

226-7. The principle of the lever 343 

228-9. Rotatory motion 347 

230. A line without mass cannot be moved . . . . .352 

231-5. The parallelogram of forces ...... 354 

236. Difficulty of analysis 364 

237. Suppressed. 

238-9. The Taylorian theorem . . . . . .367 

r. n.]. TABLE OF CONTENTS. xxi 




240. Premisses must be true in order to prove a conclusion . . i 

241. And must not covertly involve the conclusion ... 2 

242. Preposterous Reasoning confuses the principiatum as causa 

cognoscendi with the principium as causa essendi . . 3 

243. Ambiguity of middle term mostly due to the confusion of a 

relative with an absolute truth ..... 4 

244. Illustration of the above from moral precepts, all of which 

have their exceptions 5 

245. As have also mechanical formulae, which become unmeaning, 

when pushed to extremes ...... 7 

246. Fallacies of too wide or too narrow definition . . .10 

247. Fallacy of incomplete explanation illustrated by the popular 

idea that lapse of time destroys motion . . . .10 

248. Incomplete disjunction the cause of much philosophical and 

other onesidedness . . . . . . . .12 

249. The fallacy in Zeno's paradoxes about reality of motion . 14 

250. Examples of classical dilemmas stated and explained . . 17 



251. Inductive methods are based on results of deductive Logic . 22 

252. Connexions of elements revealed in sensible experience are 

mostly impure ........ 23 

253. The universality of a pure connexion or its character as a 

law of nature guaranteed by the law of Identity . . 24 

254. The raw matter of Inductions consists not of passive im- 

pressions but of perceptions already articulated by 
thought as subject and predicate and ranged under 
general conceptions . . . . . . .25 

255. They are so ranged by an incomplete analogy, based on a 

distinction of essential from non-essential remarks, which 
logical theory cannot assist . . . . . .28 

256. In reaching universal inductions we must argue ad subaltern- 

antem 31 

257. The truths of Geometry are universal because the diagram is 

used as a symbol only of our conception . . 33 

258. The highest inductions not categorical but hypothetical 

judgments 35 



1>$9. Terms which are exclusively cause and effect of each other 

are related as ground and consequent . . . -37 

260. Experiment merely subsidiary to observation and rft,s no 

peculiar virtue of its own 38 

261. Typical cases of the relation in which two phenomena C and 

E may stand to one another ...... 40 

(t) C and E Co-Exist always. 

(2) C and E frequently concur. 

(3) Absence of C not involving absence of E. Criticism 

of the canon ' cessante causa cessat et effectus.' 

(4) Presence of C not involving presence of E. Differ- 

ence of relation of cause and effect and of ground 
and consequent. 

(5) Absence of C involving absence of E. 

(6) Presence of E involving presence of C. Criticism of 

Newton's canon ' effectuum naturalium ejusdem 
generis eaedem sunt causae.' 

(7) Absence of E involving absence of C. 

262. Whether the phenomenon C is or only contains the cause of 

E can only be decided by analysis of both into their 
elements and observation of which elements of the one 
involve which elements of the other. Typical examples 
of such analysis . . . . . . . -52 

263. The exact nature of the causal nexus inferred from any of 

the above relations to exist between C and E can only be 
apprehended by observation of the quantitative changes 
they cause in one another. Examples of such quantita- 
tive correspondences ....... 60 



264. Science not content with discovering a mere connexion 

between two phenomena seeks to know the law of this 
connexion ......... 67 

265. Laws of nature are universal hypothetical judgments and not 

assertions of universal matters-of-fact .... 68 

266. A law expresses an objective and intelligible connexion of 

phenomena, a rule is a mere subjective method of thought 71 

267. The ultimate criterion of sense-perception to be found in 

sense itself 73 

268. Facts as they appear are not only relative to one another 



but to the standpoint of the observer, and must therefore 

be grasped as projections of ulterior and truer facts 76 

269. A lav always transcends the given, being an extension to 

cases not given of what holds good within the given '. A 
truly universal law is not a demonstrable truth . . 79 

270. Laws based on statistics are mostly partial truths . . 84 

271. The law which prima facie best fits in with observed facts 

need not therefore be the truest expression of their inter- 
connexion ......... 85 

272. Simplicity no guarantee for the truth of a law. The simplest 

law only preferable where it is the sole conceivable one . 87 

273. A postulate lays down the conditions under which alone the 

.given appearance is conceivable. An hypothesis is a 
suggestion of conceivable facts fulfilling the demands of 
the postulate and so explaining the appearance. A fiction 
views the given as an approximate realisation of a known 
law, in the absence of a known law to which it can be 
simply referred 90 

274. Rules for framing of hypotheses not to be laid down before- 

hand, but none to be rejected because beyond reach of 
refutation if false ........ 94 

275. Hypotheses must satisfy their postulates and supply the 

conditions of the appearances to be explained ... 97 

276. An old hypothesis not to be hastily set aside but modified to 

suit the new and discrepant facts ..... 99 

277. An hypothesis must limit itself to asserting what is possible, 

i.e. what can be conceived or pictured as matter-of-fact . 101 



278. In determining facts which transcend the immediate im- 

pression we must be guided by probability . . .104 

279. In view of the complexity of things a principle of ex- 

planation must not be too simple and abstract . .105 

280. And on the other hand it must involve as few presuppositions 

as possible. Positive evidence preferable to negative . 107 

281. The mathematical determination of chances assumes that 

they are all equally possible, but that one of them must 
occur 109 



2if2. (i) Mathematical chance no positive prediction of events. 

It measures our expectation of their occurrence - . 113 

(2) Jheory of composite chances. 

(3) dependent chances. 

(4) probability of alternative causes. 

(5) probability of an event's recurrence. 

(6) mathematical expectation. 

(7) moral expectation. 

283. Calculus of chances not only presupposes the laws of all 

calculation such as law of Identity and doctrine of dis- 
junctive judgment, but also an ordered universe of inter- 
dependent events 127 

284. Mathematical chance is our subjective expectation of vin 

event, and not a permanent property thereof. The 
resulting chance improbable only as compared with the 
sum of its alternatives, not as compared with any one of 
them . . . . . . . . . .130 

285. Success of attempts made to test by experiment the calculus 

of chances 133 

286. Such successful results not fraught with intelligible necessity, 

but the result of constant conditions operating among 
vaiiable ones, which in the long run neutralise each other 135 

287. Use of the calculus in cases where constant and variable 

causes of an often repeated event are unknown. Nature 

of so-called statistical laws . . . . . .139 

288. Use of the calculus in determining the probable accuracy of 

our observations of magnitude. The method of the least 
squares .......... 142 



289. Conditions presupposed by a logical treatment of the problem 

of expressing a collective will ...... i 

290. Defects of absolute majority . . . . . 149 

291. The weight of votes. A majority of majorities may be a 

minority of the whole constituency . . , 149 

292. Voting so as to express intensities of Volition . . .153 

293. Election by elimination 157 

294. When order of putting proposals to the vote is important . 159 

295. Rejection of innovations as stick. ' Order of the day . .161 

296. Amendments and substantive motion. Order of putting 

proposals to the vote . . . . . . .162 



On Knowledge (Methodology). 


2!) 7. Analytic and Synthetic methods practically inseparable 166 

298. Correspond respectively to Investigation and Exposition ; are 

more general than ' methods ' of applied Logic . . 1 69 
21)9. But applied Logic, like common thought, rests on untested 

bases J 7 

300. And so does science as we have it . . . T 7 T 

301. Methodology however as treatment of Knowledge is enquiry 

into sources of certainty . . . . . *73 



302. Scepticism presupposes Truth and Knowledge . . .176 

303. But doubts whether our Knowledge is Truth. Descartes . 179 

304. This doubt involves the assumption of a world of things 

which our thought should copy 182 

305. But any decision postulates the competence of thought . 184 

306. Which can only be guided by conceptions in our minds . 185 

307. Our delusion could only be revealed by fresh knowledge . 187 

308. Which must be related to the old. Things are not know- 

ledge of things . . . .189 

309. That Things may not be what they seem, as a mere general 

doubt, is self-contradictory *9 2 

310. Sceptical arguments in Sextus Empiiicus . . . . *93 

311. They involve the above difficulties . . . * .196 

312. Error in ' we only know phenomena ' *9 8 



313. Genesis of Plato's doctrine ot ' Ideas ' aoo 

314. The Ideas as Universal conceptions 202 

315. Possible knowledge of Ideas apart from question of Things . 204 

316. Distinction between Existence, Occurrence, Validity . . 206 

317. Confusion of Existence and Validity in case of the Ideas . 210 

318. Ideas in what sense eternal, And independent of things . 211 



KJ19. Aristotle on the Ideas. His universal too is ovaia . .214 
328. Modern counterparts of the Ideas. Validity a difficult 

notion . . . . . . . . . .216 

321. * Ideas impart no motion' criticised; importance of Judg- 
ments 218 



322. Judging of knowledge by our notions of its origin an illusion 223 

323. Attempt to find a starting-point for knowledge. 'Cogito, 

ergo sum' % . 226 

324. Innate Ideas; but are they true ? . . . . .229 

325. Action of one thing on another implies Spontaneity in order 

to Receptivity 231 

326. Nature of mind is contributory in #// elements of knowledge 232 

327. Both in simple Perception and in such ideas as that of causal 

connexion 234 

328. External reality must be criticised on ground of knowledge . 236 

329. Universality and Necessity as marks of a priori knowledge . 239 

330. Universal validity not derivable from repeated perceptions 

alone 241 

331. There may be spurious self-evidence* which is tested by 

thinking the contradictory 243 

332. Use of psychological analysis in establishing first principles 246 

333. Even modern Psychology hardly helps Logic . . . 248 



334. Thought must have some Real significance . . . -252 

335. Comparison and distinction as acts resulting in Relations . 254 

336. Thought is symbolic and discursive . . . . -256 

337. How can a relation of ideas be objective .... 259 

338. Only as independent of individual mind. The case of Things 260 

339. A universal cannot be realised, but has objective validity . 264 

340. Nominalism and Realism confuse Existence and Validity . 267 

341. The Reality of general notions is only validity . . . 268 
342 Conception not akin to object in structure, but in net result . 270 

343. Degrees of subjectivity in kinds of Judgment . . . 273 

344. Subjective character of Syllogism and Induction . . .276 

345. Terms antithetic to ' Subjective* and ' Formal ' . . . 279 





346. The world of Knowledge and the world of Things . . 283 

347. ' Actual Reality ' ; adequacy of Judgments to it . . .286 
J48. Applicability of thought to the course of events involves 

(i) Some given reality, which thought cannot create . 288 

349. (2) The Universality of Law in the Real world ; ultimately a 

matter of faith ........ 290 

350. And (3) synthetic judgments a priori t as basis of knowledge 

of particular laws . . . . . . . .294 

351. Hume's restriction of judgment destroys #// judgment . . 295 

352. Mathematical reasoning is not covered by the Law of Identity 297 

353. Illustration by Kant's arithmetical instance . . . .299 

354. And by his geometrical instance ..... 303 

355. Meaning and value of apprehension a priori . . . 305 

356. Self-evidence of universal Truths ..... 307 

357. Intuition is opposed to discursive thought means immediate 

apprehension ......... 309 

358. Self-evident Truths require to be discovered by help of 

analysis 311 

359. Pure Mechanics in what sense a priori . . . .313 

360. Gradual formation of pure ideas of Motion and Mass . . 316 

361. Mechanical principles, like those of Arithmetic and Geometry, 

at once identical and synthetic . . . . .319 

362. In higher Mechanics, Proof is one thing, and the Ratio legis 

another 323 

363. Analytical Knowledge as the ideal, means the simplest 

synthetical knowledge 325 

364. The simplest ultimate Truth need not be a mere datum of 

experience, though it must be Synthetic .... 327 

365. A synthetic yet necessary development the supreme goal of 

science 329 


INDEX . 333 



I. AT almost every moment of our waking life our senses 
are giving rise to various ideas, simultaneous or immediately 
successive. Among these ideas there are many which have 
a right thus to meet in our consciousness, because in the 
reality from which they spring their occasioning causes 
always accompany or follow one another ; others are found 
together in us merely because, within the external world to 
whose influence we are accessible, their causes were as 
a fact simultaneous though not so inwardly connected as to 
ensure their similar combination in every recurring instance. 
1'his mixture of coherent with merely coincident ideas is 
repeated, according to a law which we derive from self- 
observation, by the current of memory. As soon as any 
idea is revivified in consciousness, it reawakens the others 
which have once accompanied or succeeded it, whether the 
previous connexion was due to a coherence in the matter of 
the ideas, or to the mere simultaneity of otherwise uncon- 
nected irritants. It is upon the first fact, the recovery 
of what is coherent, that our hope of arriving at knowledge 
is based : the second, the ease with which coincident ele- 
ments hang together and push one another into conscious- 



ness, is the source of error, beginning with that distraction 
wi.ich hinders our thoughts from following the connexion 
of thinps. e 

II. The ever-changing whole of processes which results 
from this peculiarity of our psychical life is what we call 
the current of ideas. If it were in our power to observfe 
this whole with omniscience, we should discover in every 
instance of it, in the sober course of waking thought, in the 
dreams of sleep, in the delirium of disease, a necessary 
connexion between its members. The application of uni- 
versal laws, which hold good of all souls alike % to the 
particular conditions which are found to vary in each single 
instance, would exhibit the course of these inner processes 
in the light of an inevitable result. If we knew the per- 
manent characteristics of a single particular soul, if we had 
a view of the form and content of its whole current of ideas 
up to the present time, then, the moment it had produced 
a first and a second idea on occasion of external irritants, 
we should be able to predict on the basis of those universal 
laws what its third and fourth idea in the next moment 
must be. But in any other soul, whose nature, past history, 
and present condition were different, the same first and 
second idea, developed at this moment by a similar external 
irritant, would lead with a similar necessity in the next 
moment to an entirely different continuation. An investiga- 
tion of the subject would therefore have to recognise that 
any given current of ideas was necessary for that particular 
soul and under those particular conditions ; but it would 
not discover any mode of connexion between ideas which 
was universally valid for all souls. And just because, 
under their respective conditions, every such series of ideas 
hangs together by the same necessity and law as every 
other, there would be no ground for making any such 
distinction of value as that between truth and untruth, 
which would place one group in opposition to all the 


III. Universal validity and truth are the two prerogative? 
which even ordinary language ascribes and confines . co 
those connexions of ideas which thought alone is supposed 
to establish. Truth is familiarly defined as the agreement 
of ideas and their combinations with their object and its 
relations. There may be objections to this form of ex- 
pression, which this is not the place to consider ; but it will 
be innocuous if we modify it and say, that connexions 
of ideas are true when they follow such relations in the 
matter of the ideas as are identical for all consciousness, 
and not such merely empirical* coincidence of impressions 
as takes one form in one consciousness, another in another. 
Now our ideas are excited in the first instance by external 
influences, and this leads us to regard thought as a reaction 
of the mind upon the material supplied by those influences 
and by the results of their interaction already referred to. 
The thinking mind is not content to receive and acquiesce 
in its ideas as they were originally combined by casual 
coincidence or as they are recombined in the memory : it 
sifts them, and where they have come together merely in 
this way it does away with their coexistence : but where 
there is a material coherence between them, it not only 
leaves them together but combines them anew, this time 
however in a form which adds to the fact of their recon- 
nexion a consciousness of the ground of their coherence. 

IV. I will connect the indispensable explanation of what 
I have just said with the elucidation of some obvious 
objections. It is not without a purpose, which I admit, 
that while I have represented the rest of the current of our 
ideas as a series of events, which happen in us and to us 
according to universal laws of our nature, I have represented 
thought as an activity which our mind exercises. There 
have been persons who doubted whether this opposition 
has any real significance, either in itself or in relation to 
thought; whether everything that we are in the habit of 
calling activity is not rather one amongst the events which 

B 2 


pimply take place in us. So wide a question does not of 
cburse admit of being decided here : if therefore I hold to 
the significance of the opposition, and expressly describe 
thought as an activity, this must be regarded as a presupposi- 
tion which awaits proof elsewhere, but is at present open 
to dispute. It is necessary for the connexion of the whole" 
to which I wish this view of thought to serve as an intro- 
duction; and it seems to me to be permissible, because, 
while it will determine decisively the general colour of my 
exposition, it will not alter unnaturally the internal relations 
of its subject-matter. . 

V. It is more profitable to meet another form of the 
same objection, which allows the general validity of the 
opposition in question, but holds that there is no occasion 
to apply it here. The connexion of the coherent (it is 
said), that is to say, Truth, is brought about in the same 
way, only not quite so soon, as the erroneous conjunctions 
of the casually coincident. The course of things itself 
ensures that those events which are inwardly connected 
exercise their combined effect upon us with incomparably 
greater frequency than those which have no inward bond, 
but are variously thrown together by chance. Owing to 
this more frequent repetition the connexion of what is 
coherent becomes fixed in us, while that of the merely 
coincident is loosened and disturbed by its want of urji- 
formity. In this way the separation of the coherent from 
the incoherent, which we thought it necessary to ascribe to 
a special reaction of the mind, is effected by the current of 
ideas itself; and thus brutes, like men, acquire the mass of 
well-grounded information which regulates the daily life of 
both. It would be superfluous to point out that this 
account is perfectly correct if it purport to be no more than 
a history of the acquisition just mentioned ; but I think it 
can be shown that this acquisition is just what neither 
characterises nor exhausts the specific work of thought. 

VI. There is a common opinion which reserves the 


faculty of thought to man and denies it to brutes. With 
out seriously deciding for or against this view, I will use it 
for the convenience of my explanation. In the s$ul of a 
brute, which on this theory would be confined to a mere 
current of ideas, the first impression of a tree in leaf would 
only produce a collective image ; there would be no power 
or even impulse to seek for any special coherence between 
its parts. Winter strips the tree of its leaves, and on a 
second observation the brute finds only a part of the former 
collective image, which tries to reproduce the idea of the 
rest, bitf is hindered by the present appearance. When the 
return of summer restores the old state of things, the 
renewed image of the whole tree in leaf may not, it is true, 
have the simple unquestioning unity of the first observation ; 
the recollection of the second intervenes, and separates it 
into the part which remained and the part which changed. 
I do not think we can say what precisely would take place 
in the soul of the brute under these circumstances ; but 
even if we ascribe to it the additional faculty of comparing 
and surveying the current of its ideas and expressing the 
result, the expression could not say more than the fact that 
two observations were at one time together, at another not. 
Now it is true that the man, when he gives the name of 
leafy and leafless tree to the same observed objects, is only 
expressing the same facts ; but the apprehension of the 
facts, which is indicated by these habitual forms of speech, 
involves a mental operation of quite a different kind. The 
name of the tree, to which he adds and from which he 
takes away the descriptive epithet, signifies to him, not 
merely a permanent as opposed to a changeable part in his 
observation, but the thing in its dependence on itself and 
in opposition to its property. The effect of bringing the 
tree and its leaves under this point of view is, that the 
relationship of thing and property appears as the justifica- 
tion both for separating and for combining these ideas, and 
thus the fact of their coexistence or non-coexistence in our 


5gnsciousness is referred to the real condition upon which 
their coherence or non-coherence at the moment depends. 

The rsame consideration may be extended to other 
instances. In the soul of the dog the renewed sight of the 
raised stick recalls the idea of his previous pain : the man, 
when he makes the judgment, ' the blow huFts,' does no 
merely express the fact of connexion between the two 
occurrences, but justifies it. For in representing the blow 
as the subject from which the pain proceeds, he clearly 
exhibits the general relationship of cause and effect as the 
ground, not of the mere coexistence of the two ideals in us, 
but of their right and obligation to follow one another. 
Lastly, the expectation of pain in the dog may be accom- 
panied by the recollection that by running away, to which he 
was led before by an involuntary instinct, the pain is 
diminished ; and this fresh conjunction of ideas will doubt- 
less make him repeat the salutary operation as surely as if 
he reflected and concluded that threatening blows are pre- 
vented by distance, that a blow threatens him, and that 
therefore he must run away. But the man who in a similar 
or more serious case actually frames such a conclusion, 
performs an entirely different mental operation ; in express- 
ing a universal truth in the major premiss, and bringing a 
particular instance under it in the minor, he not merely 
repeats the fact of that salutary connexion between ideas 
and expectations by which the brute is affected, but he 
justifies it by an appeal to the dependence of the particular 
upon its universal. 

VII. These examples, which embrace the familiar forms 
of thought, concept, judgment, and syllogism, will I think 
have made sufficiently clear what is the surplus of work 
performed by thought over and above the mere current of 
ideas ; it always consists in adding to the reproduction or 
severance of a connexion in ideas the accessory notion of a 
ground for their coherence or non-coherence. The value 
of this work remains entirely the same, whatever opinion we 


may hold of its genesis : if we preferred to regard it, not as 
the outcome of a special activity, but only as a finer procKot 
of the mere current of ideas operating under favorable 
circumstances, we should then confine the name of thought 
to that particular stage of development in the current at 
which it gives birth to this new achievement. The pecu- 
liarity of thought, then, which will govern the whole of our 
subsequent exposition, lies, not in the mere correspondence 
of our apprehension with fact, but in the production of 
those accessory and justificatory notions which condition 
the form of our apprehension. We do not deny that, apart 
from thought, the mere current of ideas in the brute gives 
rise to many useful combinations of impressions, correct 
expectations, seasonable reactions ; on the contrary, we 
admit that much even of what the man calls his thought is 
really nothing but the play of mutually productive ideas. 
And yet perhaps there is still some difference here. The 
sudden inspirations which enable us to make a decision in 
a moment, the rapid survey which arranges a complicated 
material in almost less time than would seem sufficient for 
the bare observation of its parts, the invention of the artist 
which remains unconscious of the grounds by which it is 
impelled, all these seem to us to be effects, not of a current 
of ideas which has not yet become thought, but of abbre- 
viated thought. In the cases where these surprising 
operations are successful, they are so because mature 
thought has already in other cases developed into full-grown 
habits those accessory notions, which bring the impressions 
under universal principles of coherence ; and this, like all 
other accomplishments which have acquired the ease of a 
second nature, has behind it a forgotten time of laborious 

VIII. In the examples which I have employed, the 
accessory notions, by which we justify the connexions of 
ideas, obviously coincided with certain presuppositions 
about the connexions of the real with which we cannot 


dispense. Without the opposition of things and their 
pi ^oerties by which the whole matter of perception is 
articulated, without the assumption of a succession of 
effects from causes, and without the determining power 
of the universal over the particular, we could have no 
apprehension whatever of the reality which surrounds us: 
From this point of view, then, it seems a self-evident 
proposition that the forms of thought and the accessory 
notions which give them vitality are immediate copies of 
the universal forms of being and its connexions, and this 
real validity of thought and its operations has, in fact, been 
frequently maintained. The opposite view to this, which 
as its exact counterpart we might expect to find, has never 
been put forward so unreservedly. To the unprejudiced 
mind it is too natural to regard thought as a means of com- 
prehending the real, and any interest in the scientific 
investigation of its processes is too dependent upon this 
presupposition for any one to assert the merely formal 
validity of all logical activity in the sense of denying all 
relation between it and the nature of being. Those, there- 
fore, who have regarded the forms and laws of thought as 
being primarily peculiar results of our mental organisation, 
have not wholly excluded their correspondence with the 
essence of things ; they have only denied the off-hand view 
which would make them immediate copies of the forms 
of being. 

IX. In regard to this much debated question an intro- 
duction can only take up a provisional position. We shall 
certainly be right to confine our attention at starting to 
what is already clear, and to leave for a later stage the 
decision of uncertainties. Let us then go no further than 
the natural presupposition which regards thought as a means 
to knowledge. Now a tool must fulfil two conditions, it 
must fit the thing and it must fit the hand. It must fit the 
thing ; that is, it must be so constructed as to approach, 
reach, and get hold of, the objects which it is to work upon, 


and find in them a point from which to operate; this 
requirement is satisfied in the case of thought if we acKat 
that its forms and laws are no mere singularities y of our 
mental organisation, but that, taken as they are, they show 
a constant and regular adaptation to reality. If, again, 
a tool is to fit the hand, it must have such other structural 
properties as make it easy to grasp, hold, and move, having 
regard to the power, attitude, and position of the person 
who is to use it ; and in the case of thought this second 
indispensable requirement limits the scope of the previous 
admissipn. Only a mind which stood at the centre of the 
real world, not outside individual things but penetrating 
them with its presence, could command such a view of 
reality as left nothing to look for, and was therefore the 
perfect image of it in its own being and activity. But the 
human mind, with which alone we are here concerned, does 
not thus stand at the centre of things, but has a modest 
position somewhere in the extreme ramifications of reality. 
Compelled, as it is, to collect its knowledge piece-meal 
by experiences which relate immediately to only a small 
fragment of the whole, and thence to advance cautiously to 
the apprehension of what lies beyond its horizon, it has 
probably to make a number of circuits, which are im- 
material to the truth which it is seeking, but to itself in the 
search are indispensable. However much, then, we may 
presuppose an original reference of the forms of thought to 
that nature of things which is the goal of knowledge, we 
must be prepared to find in them many elements which do 
not directly reproduce the actual reality to the knowledge 
of which they are to lead us : indeed there is always the 
possibility that a very large part of our efforts of thought 
may only be like a scaffolding, which does not belong to the 
permanent form of the building which it helped to raise, 
but on the contrary must be taken down again to allow the 
full view of its result. It is enough to have thus raised 
a preliminary expectation, with which we wish our subject 


to be met; any more definite decision as to the limits 
\vHch separate the formal validity of our thought from its 
real significance must await the further course of our en- 

X. I have purposely avoided postponing those enquiries 
by discussions which seem to me to encumber unjustifiably 
the approach to logic. What particular tone of mind is 
required for successful thinking, how the attention is to 
be kept up, distraction avoided, torpidity stimulated, pre- 
cipitation checked, all these are questions which no more 
belong to the field of logic than do enquiries about the 
origin of our sense-impressions and the conditions under 
which consciousness in general and conscious activity is 
possible. We may presuppose the existence of all these 
things, of perceptions, ideas, and their connexion according 
to the laws of a psychical mechanism, but logic only begins 
with the conviction that the matter cannot end here ; the 
conviction, that between the combinations of ideas, however 
they may have originated, there is a difference of truth and 
untruth, and that there are forms to which these combina- 
tions ought to answer and laws which they ought to obey. 
It is true that we may attempt by a psychological investiga- 
tion to explain the origin of this authoritative consciousness 
itself; but the only standard by which the correctness 
of our results could be measured would be one set up 
by the very consciousness to be investigated. The first 
thing, then, that has to be ascertained is, what the con- 
tents of this authoritative conviction are ; the history of its 
growth can only have the second place, and even then must 
conform to requirements of its own imposing. 

XI. Having now said all that seemed necessary by way 
of introduction to my exposition, I will add a preliminary 
survey of its order. The examples which we have hitherto 
employed lead naturally to a first principal part, which, 
under the name of pure or formal logic, is devoted to 
thought in general and those universal forms* and principles 


of thought which hold good everywhere, both in judging of 
reality and in weighing possibility, irrespective of any differ- 
ence in the objects. We have only to mention concept, 
judgment, syllogism, to see how naturally these forms 
exhibit themselves as different stages of one and the same 
activity; and in treating of pure logic I shall endeavour 
to emphasise this thread of connexion somewhat more 
strongly than is usually done. The various forms of 
thought will be arranged in an ascending series, in which 
each higher member attempts to make good a defect 
in the preceding one, due to its failure to satisfy, in regard 
to its own particular problem, the general impulse of 
thought to reduce coincidence to coherence. This series 
will advance from the simplest formation of single impres- 
sions to the conception of the universal order in which 
this general impulse would lead us, if it were possible, 
to comprehend the world. 

XII. Pure logic itself will show and explain that the 
forms of concept, judgment, and syllogism are to be con- 
sidered primarily as ideal forms, which give to the matter of 
our ideas, if we succeed in arranging it under them, its true 
logical setting. But the different peculiarities of different 
objects offer resistance to this arrangement ; it is not clear 
of itself what sum of matter has a claim to form a deter- 
minate concept and be opposed to another, or which 
predicate belongs universally to which subject, or how the 
universal law for the arrangement of a manifold material is 
to be discovered. Applied logic is concerned with those 
methods of investigation which obviate these defects. It 
considers hindrances and the devices by which they may be 
overcome ; and it must therefore sacrifice the love of 
systematisation to considerations of utility, and select what 
the experience of science has so far shown to be important 
and fruitful. The boundlessness of the field of observation 
unfortunately makes it impossible to exhibit as completely 
as could be wished this most brilliant part of logic, which 


*he inventive genius of modern times has made peculiarly 
its nwn. 

XK^ The third part will be devoted to knowledge, that 
is, to the question which our introduction touched without 
answering, the question how far the most complete structure 
of thought which all the means of pure and applied logic 
enable us to rear, can claim to be an adequate account 
of that which we seem compelled to assume as the object 
and occasion oF our ideas. The currency in ordinary 
minds of this opposition between the object of our 
knowledge and our knowledge of that object makes me 
employ it without hesitation to describe in a preliminary 
way the subject of this third section ; it may be left to the 
section itself to disclose the difficulties which this apparently 
simple antithesis involves, and to determine accordingly 
the more precise limits of the problems with which it has 
to deal. 


The Theory of the Concept. 
A. The formation of impressions into ideas. 

1. IT is in relations within a manifold that the operations 
of thought usually show themselves to us, and we might 
therefore expect to have to look for the most original of its 
acts in some simplest form of connexion between two ideas. 
A slight reflexion, however, suggests to us to go a step 
further back. It is easy to make a heap out of nothing but 
round stones, if it is indifferent how they lie; but if a 
structure of regular shape is to be built, the stones must be 
already so formed that their surfaces will fit firmly together. 
We must expect the same in the case before us. As mere 
internal movements, the states which follow external irritants 
may exist side by side in us without further preparation, and 
act upon each other as the general laws of our psychical life 
allow or enjoin. But if they are to admit of combination 
in the definite form of a thought, they each require some 
previous shaping to make them into logical building-stones 
and to convert them from impressions into ideas. Nothing 
is really more familiar to us than this first operation of 
thought ; the only reason why we usually overlook it is that 
in the language which we inherit it is already carried out, 
and it seerns therefore to belong to the self-evident pre- 
suppositions of thought, not to its own specific work. 


2. That which takes place in us immediately under the 
iriVuence of an external stimulus, the sensation or the 
feeling is in itself nothing but a state of our consciousness, 
a mood of ourselves. We do not always succeed in 
naming, and so making communicable to others, the 
manner in which we are thus affected; sometimes the 
formless interjection, the exclamation, is the only way we 
can find, though with no certainty of being understood, to 
give sound to what cannot be said. But in the more 
favorable cases, where we have succeeded in creating a 
name, what exactly is it which this creation effects and 
indicates? It is just what we are here looking for, the 
conversion of an impression into an idea. As soon as we 
give the name of green or red to the different movements 
which waves of light produce through our eyes, we have 
separated something before unseparated, our sensitive act 
from the sensible matter to which it refers. This matter 
we now present to ourselves, no longer as a condition 
which we undergo, but as a something which has its being 
and its meaning in itself, and which continues to be what 
it is and to mean what it means whether we are conscious 
of it or not. It is easy to see here the necessary beginning 
of that activity which we above appropriated to thought as 
such : it has not yet got so far as converting coexistence 
into coherence, it has first to perform the previous task .of 
investing each single impression with an independent 
validity, without which the later opposition of their real 
coherence to mere coexistence could not be made in any 
intelligible sense. 

3. We may describe this first operation of thought as the 
beginning of an obj edification of the subjective ; and I take 
advantage of this expression to guard against a misunder- 
standing and so illustrate the simple meaning of what I 
have said above. It is not objectivity in the sense of some 
sort of real existence which would subsist though nobody 
had the thought of it, that, by the logical act of creating a 


name, is accorded to the subject-matter to which that act 
gives rise. The true meaning of the first act of though/is 
best exemplified by those languages which have maintained 
the use of the article. The article, which had everywhere 
originally the value of a demonstrative pronoun, marks the 
word which it accompanies as the name of something to 
which we point ; and what we point to is something which 
admits of being observed by another person as well as by 
ourselves. This can be done most easily with things which 
have an actual position in space between the speakers ; but 
developed language makes an object of any other matter 
of thought in the same way. Such objectivity, therefore 
(which in these cases also is indicated by the article), does 
not entirely coincide with the reality which belongs to 
things as such ; it is only the fact of their claiming such 
a reality, on the ground of the distinctive peculiarity of 
their real nature, which language has met and expressed 
in their names. When we speak of 1 'the tooth-ache/ 
'the day,' 'the franchise,' we do not imply that they could 
exist if there were no person to feel, to see, to enjoy them, 
respectively. Still less when we talk of 'the adverb' or 
'the conjunction,' do we mean to indicate by the article 
that the subject-matter described by these words has any 
sort of existence outside thought. We only mean that 
certain special forms of resistance and tension, which we 
feel in the course of our ideas, are not only peculiarities of 
our own state and inseparable from it, but that they 
depend upon relations inherent in the matter of various 
ideas, which every one who thinks those ideas will find in 
them just as we do. 

The logical objectification, then, which the creation of 
a name implies, does not give an external reality to the 

1 [The instances in the text are der Schmerz, die Hclligkeit, die 
Freiheit, but none of the equivalents are used in the required sense with 
the article in English. The same applies to the instances in the follow- 
ing sentence, das Zwar, das Aber, das Dennock.'] 


matter named; the common world, in which others are 
expected to recognise what we point to, is, speaking 
gene\.;]ly, only the world of thought ; what we *do here is 
to ascribe to it the first trace of an existence of its own and 
an inward order which is the same for all thinking beings 
and independent of them : it is quite indifferent whether 
certain parts of this world of thought indicate something 
which has besides an independent reality outside the 
thinking minds, or whether all that it contains exists only 
in the thoughts of those who think it, but with equal validity 
for them all. 

4. But the objectification of the matter so first constituted 
is not the whole of this first act of thought ; consciousness 
cannot simply present the matter to itself, it can only do so 
by giving it a definite position ; it cannot simply distinguish 
it from an emotional mood of its own, without accrediting 
it with some other sort of existence instead of that which 
belonged to it as such a mood. The meaning of this 
requirement (for I admit that my expression of it is not 
immediately clear) is most simply shown by the way in 
which language actually satisfies it. It is only in the 
interjection, which is not a name of definite content, that 
language retains the formlessness which belongs to it as the 
mere expression of excitement ; the rest of its stock of 
words is articulated in the definite forms of substantives, 
adjectives, verbs, and the familiar parts of speech in general. 
And it is hardly necessary to insist that the various char- 
acters thus impressed by language upon its material are the 
indispensable condition of the later operations of thought ; 
it is obvious that neither the combination of marks into 
the concept, of concepts into the judgment, or of 
judgments into the syllogism would be possible, if the 
matter of every idea were equally formless or appre- 
hended in the same form, if some of them were not 
substantival and did not express fixed and independent 
points of attachment for others which are adjectival, or if 


others again were not verbal, exhibiting the fluid relations 
which serve to bring one thing into connexion with another. 
I do not ' think it advisable to separate this pabular 
conformation of the matter of ideas, as a second act of 
thought, from the first act, to which we ascribed its ob- 
jectification ; I prefer to comprise the primary activity of 
thought in a single operation, which may be indifferently 
represented as that of giving to the matter of ideas one 
of these logical forms by making it objective for conscious- 
ness, or as that of making it objective by giving it one 
of these^forms. 

5. The three parts of speech which I have noticed 
remind us inevitably of three concepts which are indis- 
pensable for our judgment of reality. It is impossible to 
have even an expressible idea of the world of perception, 
without thinking of things in it as fixed points which serve 
to support a number of dependent properties, and are 
connected together by the changing play of events. If 
metaphysic is the investigation, not of the thinkable in 
general, but of the real or that which is to be recognised 
as such, these concepts of thing, property, and event are 
metaphysical concepts; not perhaps such as metaphysic 
would finally allow to stand without modification, but cer- 
tainly such as at its outset purport to represent immediately 
the proper essence and articulation of what is. 

It would seem at first sight that the logical forms of 
substantivity, adjectivity, and verbality coincide with these 
concepts : but a second view shows the same difference 
between the two series as that which separated the logical 
objectification of an idea from external reality. Nothing 
passes with us for a thing or a substance which has not 
reality outside us and permanence in time, producing 
changes in something else and capable of undergoing 
changes itself; but we apprehend as substantives not only 
things but their properties; as substantives we speak of 
* change/ * occurrence,' even of * nothing/ and so in in- 



numerable cases of that which has no existence at all or 
no\^ except in dependence on something else. 9 Thus the 
substantival form invests its content, relatively to the future 
predicates to which it is to serve as subject, with only 
the same priority and independence as belong to a thing 
in contrast with its properties, conditions, and effects, but 
by no means with that concrete and independent reality 
and activity which place a thing above a mere object of 

Verbs, again, express most frequently an event which 
as a fact takes place in time ; but when we say that things 
'are/ or 'are at rest,' or that one ' conditions' or 'equals 7 
another, it is clear that the verbal form too does not 
universally give to its content the meaning of an event, 
but only finds it there usually. In order to conceive fully 
the sense of such verbs as we have just instanced, we 
have to connect several distinct contents together by a 
movement of thought, and this movement though it implies 
time for its execution, is, as regards its meaning and in- 
tention, quite independent of time. In a word, the general 
sense of the verbal form is not an event, but a relation 
between several related points ; and this relation may just 
as well occur between contents which are out of time and 
coexist only in thought, as between those which belong 
to reality and are accessible to temporal change. 

Lastly, while it is true that radical adjectives, such as 
'blue' and 'sweet,' express primarily what appears to our 
first apprehension as a real property of things, every de- 
veloped language knows words like 'doubtful/ 'parallel,' 
'allowable,' which, as the least reflexion shows, can no 
longer mean in the same simple sense as the former a 
property attaching to actual things; they are abbreviated 
and condensed expressions of the result of all sorts of 
relations, and it is only for purposes of thought that we 
represent the contents of such adjectives as related to 
those of substantives in the way in which we imagine an 


attribute to be related to its subject. Speaking generally, 
then, the Jogical import of the parts of speech is o**.iy a 
shadow of the import of these metaphysical concepts : 
it only repeats the formal characteristics which the latter 
assert of the real; but by not confining their application 
to the concrete external reality, it loses that part of their 
significance which they only possess in that application. 

6. Lastly, if we found in the forms of the parts of speech 
the most original activity of thought, we must also under- 
stand how to distinguish this from its linguistic expression. 
Now that man has come to use the language of sounds 
for the communication of his thoughts, that activity is, it 
is true, most clearly manifested in the forms of the parts 
of speech ; but in itself it is not inseparably bound up 
with the existence of language. The development of which 
the ideas of the deaf and dumb are capable, though guided 
in the first instance by those who can speak, is enough 
to show that the internal work of logic is independent of 
the possibility of linguistic expression. That work consists 
merely in the fact that we accompany the content of one 
idea with the thought of its comparative independence, 
while we think of another as requiring support, and of 
a third as a connecting link which neither subsists on 
its own account nor rests upon something else but me- 
diates between two others. No one doubts the extremely 
effective support which language gives to the development 
of thought by making the formations and transformations 
of ideas vividly objective to consciousness by means of 
sharply defined sounds and their regular changes; still, if 
some other mode of communication were natural to man 
instead of the language of sounds, the same logical asso- 
ciations would find in it a corresponding expression though 
of a different kind. And if in some languages the poverty 
of forms does not always allow these associations to take 
shape, cannot, for instance, distinguish between substantival 
and verbal construction, yet there is no doubt that the mind 

C 2 


of those who speak them maintains the logical distinctions 
wh.^e forming ideas which are vocally undistinguished. 
Whertver there is this inward articulation, there is thought; 
where it is wanting, there is no thought. For this reason 
music is not thinking ; for however manifold and delicately 
gradated are the relations of its tones, it never brings them 
into the position of substantive to verb, or into dependence 
such as that of an adjective on its noun or a genitive on 
the nominative by which it is governed. 

7. In mentioning hitherto only three out of a greater 
number of parts of speech, the three without which the 
simplest logical enunciation would be impossible, I do not 
wish to deny the value of the others. But the road which 
we have to traverse is too long to allow us to make further 
circuits into the attractive field of philological enquiry, 
circuits which, considering how thought has just been said 
to be independent of its mode of expression, must for our 
purpose remain circuits. The articulation and usage of 
language do not fully cover the work of thought. We 
shall find later that they frequently do not express the 
complete structure of the thought ; and then we have for 
the purposes of logic to supplement what is said by what 
was meant. On the other hand language possesses technical 
elements which do not depend, or only depend with various 
degrees of remoteness, upon characteristics essential to 
logic : in such cases we should not be justified in distin- 
guishing a different logical operation of thought for every 
grammatical or syntactical difference of form presented by 
language. There are not only interjections, but particles 
too, which, like the tone of the voice, hardly indicate more 
in ordinary usage than the interest which the speaker feels 
in what he is saying, and contribute nothing to its sub- 
stantial logical meaning. When language introduces the 
distinction of gender into all substantives and adjectives, 
it follows an aesthetic fancy which has no interest for logic ; 
when on the other hand it determines the gender of the 


adjective by that of its substantive, this consistency in an 
arbitrarily adopted custom points back to a logical relation- 
ship which we shall become acquainted with. When : A the 
inflexions of the verb it distinguishes the person speaking 
from the person spoken to and the third person not present, 
it emphasises an extremely important fact in a way which 
is indispensable for the living use of speech, and yet there 
is no corresponding distinction in logic proper. It is 
nothing but the same reason which justifies grammar in 
considering pronouns as a specific class of parts of speech : 
logically;, the personal pronouns must be reckoned entirely 
among substantives, with which in formal position they are 
identical; the possessive and demonstrative we have no 
ground for separating from adjectives; the relative we 
should regard as the most specifically technical element in 
language, serving only the need of methodical communica- 
tion, and based on no other logical relation than its counter- 
part the demonstrative. Numerals are treated by grammar 
as distinct parts of speech ; in the actual usage of language 
they are equivalent to adjectives, and that logically they 
belong to the latter we cannot doubt, when we remember 
that logically the form of adjectivity belongs to all charac- 
teristics of a subject-matter which are not self-dependent, 
and not only to those which attach to it in the sense of 
properties. Adverbs, lastly, stand in precisely the same 
relation to the meaning of verbs as adjectives to that of 
substantives, so that logic would have no occasion to con- 
sider them as a distinct part of speech or a peculiar form 
of the content of thought. 

Thus there would only remain prepositions and con- 
junctions to put forward such a claim, and of them I think 
we must admit that, however they may be derived linguisti- 
cally, they form an indispensable element in the world of 
our ideas. They cannot be derived from the concept of 
relation, with which at first they seem to be connected : 
whenever two members are connected by a relation, there 


is involved the thought of a certain position which those 
members occupy within the relation itself, and this position 
neea not be the same for both; on the confrary, it is 
generally different, the one embracing, containing, and 
conditioning the other. Now it will be found upon trial 
to be impossible to express this difference of value between 
the related points, without which the relation has no 
meaning, in a merely verbal form : somewhere or other 
we shall need a preposition, a conjunction, or at least 
one of the various case-forms in which many languages 
express some of these accessory notions still more shortly. 
In what linguistic form they appear, is of course quite 
indifferent to logic ; just as we oppose the nominative, as 
that which conditions, sometimes to the genitive, as that 
which is conditioned, sometimes in a different sense to the 
accusative, so, if language had produced or preserved a 
still greater wealth of cases, all prepositions would be 
superfluous, as all conjunctions would be if there were a 
similar variety of moods. This would make no change in 
the logical needs of thought; in one way or another, the 
meanings of substantives, adjectives, and verbs would have 
to be supplemented by a number of ideas, indicating, either, 
like prepositions, the position of two supposedly simple 
objects in a simple relation, or, like conjunctions, the com- 
parative position and value of two relations or judgments. 

8. If we glance at the developed structure of the world 
of our thoughts and ask what the conditions are upon 
which its construction depends, the objectification of im- 
pressions and their concomitant formation in the sense 
of the parts of speech must always appear as the most 
indispensable, and in that sense the first, of all operations 
of thought. It is certain that without it the framing of 
sentences, simple or complex, through which we express 
the work and results of our thinking, would have been 
quite impossible. But we must not be taken to mean 
that the logical spirit, at the beginning of its intellectual 


work, before it ventured a step further, performed this, 
the first of its necessary operations, on the entire matter 
of its ideus once for all. The infinitude of possible Im- 
pressions, of which every moment may bring a new one, 
would be enough to make such a task impracticable : it 
is made still more impracticable by the fact that in work- 
ing up the matter that is given to it thought is constantly 
producing new matter, and has to bring this again into the 
same logical forms of which, as applied to a simpler matter, 
it is the result. Thus it is that every developed language 
possesses, in the form of simple substantives, adjectives, 
or verBs, numerous ideas which could not have been 
framed, and cannot be fully understood, without manifold 
intellectual operations of a higher kind, without employing 
judgments and syllogisms, and even without presupposing 
systematic scientific investigation. 

This obvious reflexion has given rise to the assertion, 
that in logic the theory of judgment at least must pre- 
cede the treatment of concepts, with which it is only 
an old tradition to begin the subject. I consider this to 
be an over-hasty assertion, due partly to a confusion of the 
end of pure with that of applied logic, partly to a general 
misconception of the difference between thought and the 
mere current of ideas. For if those judgments, out of 
which the concept is said to result, are to be really judg- 
ments, they themselves can consist of nothing but com- 
binations of ideas which are no longer mere impressions ; 
every such idea must have undergone at least the simple 
formation mentioned above \ the greater part of them, 
as experiment would show, will already practically possess 
that higher logical form to which the very theory in question 
gives the name of concept. The element of truth in this 
proposed innovation reduces itself to the very simple 
thought, that in order to frame complex and manifold con- 
cepts, more especially in order to fix the limits within 
which it is worth while and justifiable to treat them as 


wholes and distinguish them from others, a great deal of 
preparatory intellectual work is necessary ; but that this 
prep*, "itory work itself may be possible, it must have 
been preceded by the conformation of simpler concepts 
out of which its own subsidiary judgments are framed. 
Without doubt, then, pure logic must place the form of 
the concept before that of the judgment : it remains for 
applied logic to tell us how, in framing determinate con- 
cepts, judgments consisting of simpler concepts may be 
turned to account. A proposal to reverse this order can 
only commend itself to those who regard thinking in 
general as merely the interaction of impressions excited 
in us from without, and overlook the reacting energy 
which makes itself felt at every point in the current of 
ideas, separating the merely coincident, combining the 
coherent, and thus already giving form to the individual 
elements of future thoughts. 

B. Position l , Distinction, and Comparison of the Matter 
of Simple Ideas. 

9. If we recognise in these first formative acts the specific 
contribution which the operative energy of thought makes 
to the whole of our intellectual world, we are easily led to 
the view that the logical spirit has certain ready-made 
modes of apprehension with which it meets the impressions 
as they come ; and this again raises the question, how it 
contrives to bring the matter of each impression under that 
particular form which is appropriate to it. But such a view 
is inadmissible, and such a question therefore has no point, 
or at any rate leads to an answer different from that 
which it expects. Thought does not stand fronting the 
impressions as they arrive with a bundle of logical forms 
in its hand, uncertain which form can be fitted to which 
impression, and therefore needing some special expedient 

1 [' Position,' as the equivalent of Setzung, is here used in the active 
sense in which it occurs, e. g. in ' composition.'] 


to discover how to pair them properly. It is the relations 
themselves, already subsisting between impressions when 
we become conscious of them, by which the act 1 '- ii of 
thought, which is never anything but reaction, is attracted ; 
and this action consists merely in interpreting relations, 
which we find existing between our passive impressions, 
into aspects of the matter of the impressions. It is not 
therefore the assignment of the proper form of each matter, 
which requires any special device of thought : in another 
point of view, however, this arrangement of a manifold 
matter in logical forms does involve a second intellectual 
operation; for no matter can have a name made for it 
unless it has been thought of as identical with itself, as 
different from others, and as comparable with others. 

10. This second operation of thought, like the first, is 
one which inherited language has already carried out for all 
those who speak it ; like the first therefore it is easily over- 
looked, and not reckoned as part of the work of the mind. 
But logical science, expressly devoted to the self-evident, 
must not treat a part of its subject as a still more self- 
evident presupposition which may be excluded from the 
proper objects of its consideration. Still, the first at any 
rate of the three heads under which we expressed this new 
operation of thought does not need a detailed explanation. 
It is at once obvious how every name, ' sweet ' or t warm,' 
* air ' or 4 light/ ' tremble ' or ' shine,' gathers up the matter 
which it indicates in some sort of coherent unity with a 
meaning of its own; it is not only (though it is most 
emphatically) matter in the substantival form that is thus 
lifted into unity with itself by the prefixed article; the 
same indicative force resides, under a different form, in the 
infinitive of the verb, and even when language has no 
distinctive expression for it, this accessory notion of 
singling out and giving position to the matter indicated 
accompanies every form of word. It may be doubted 
whether the process which we would understand by giving 


position is not already contained in the objectification which 
we represented as converting the passive impression into 
an\"^ea; and it is true that we can neither have an idea 
without thus giving position to its content, nor give it 
position in any intelligible sense without objectifying it. 
Practically therefore it is a really inseparable operation 
which we are considering from different sides ; before, we 
contrasted the presented idea to which we are related as 
presenting, with the impression by which we are simply 
affected; now, when the multiplicity of the matter pre- 
sented begins to excite our attention, we lay stress upon 
the unity and independence in virtue of which the matter 
thus singled out by attention is what it is and differs from 
everything else. 

11. By the last words I wished to convey clearly the 
close connexion in which the affirmative position given to a 
content stands with the negative exclusion of all others. 
The connexion is so close, that the terms which we are 
obliged to employ to express the simple sense of the first 
are only made perfectly clear by adding the accessory notion 
of the second. We can only explain what we mean by the 
unity of position given to a content by emphasising its 
difference from others, and saying, not only, it is what it is, 
but also, it is not what others are. The affirmation and the 
negation are one inseparable thought, and accompany in 
inseparable union every one of our ideas, even when we do 
not expressly attend to the others which are tacitly negated. 
But the accessory notion thus amalgamated with our ideas 
only determines the logical setting which we give to their 
content; it does not produce that content in the first in- 
stance. It cannot be said that we have the idea of red as 
red only when we distinguish it from blue or sweet, and 
only by so distinguishing it, and again the idea of blue as 
blue only by a similar opposition to red. There could be 
no conceivable occasion for attempting such a distinction, 
nor any possibility of succeeding in the attempt, unless 

Chap. I.] COMPARISON. 2? 

there were first a clear consciousness of what each of the 
two opposites is in itself. Without doubt the peculiar 
impression which we experience under the influe^e of 
red light will be entirely the same before we have had our 
first experience of blue light as it will be afterwards; the 
possibility of comparison and distinction which the latter 
experience gives may indeed, at any rate in a matter more 
complex than these simple colours, draw the attention to 
parts of the impressions which had been previously over- 
looked, and so make both of them more complete; but 
even in this case, which is quite outside our present con- 
sideration, the new element is not discovered by the dis- 
tinction, but by the immediate sensation of which the 
comparison was merely the occasion. It is always affirma- 
tive position therefore which makes negative distinction 
possible, while it is never the case that the act of distinction 
gives rise to the matter distinguished. Only our accessory 
notions about the matter of our ideas, only its logical 
setting, gains in definiteness by adding to the affirmation 
of itself the negation of others ; and even this gain would 
seem to me small if it went no further, and were not sup- 
plemented by that third operation of positive comparison, 
which, in the above account of this second act of thought, 
was mentioned last. , 

12. I will introduce the consideration of this third opera- 
tion, which I regard as the most essential part of the logical 
work to be here explained, by recalling a familiar fact 
which is commonly used to support other conclusions. 
Words never denote impressions as they can be experienced ; 
we can only experience or actually perceive a particular 
shade of red, a specific kind of sweetness, a definite degree 
of warmth, not the universal red, sweet, and warm of 
language. The universalisation which in these and all 
similar cases the matter of sensation has undergone, is 
commonly regarded as an unavoidable inexactness of 
language, perhaps even of the thought which language 


serves to express. Unable or not accustomed to make a 
definite name for every single impression, language (it is 
supposed) blurs the slight differences between them, and 
retains only what is immediately experienced in sensation 
as common to them all : by this reduction of its means of 
expression to a moderate number it certainly makes the 
communication of ideas possible, but diminishes propor- 
tionately the exactness of that which has to be communi- 
cated. I do not think that this view does full justice to 
the significance of the fact. 

13. First of all, to regard the universalisation in question 
as a sort of falsification of the impressions is to pass too 
lightly over the very remarkable circumstance, that in a 
number of different impressions there is something common 
which can be thought apart from their differences. This is 
by no means such a matter of course that the opposite is 
out of the question ; on the contrary, it is quite conceivable 
that every one of our impressions should be as incomparably 
different from every other as sweet actually is from warm, 
yellow from soft. The fact that the thinkable world itself 
is so constituted that this is not the case, is one which it is 
worth while to take into consideration. Nor again can I 
regard the want of exactness, which the application of the 
universal terms of language undoubtedly gives to the com- 
munication of ideas, as sheer loss. Moreover, when perfect 
exactitude is felt to be important, the shortcomings of these 
simplest products of rudimentary thought can always be 
supplemented by its more advanced activity : science has 
long taught us to measure every degree of heat, and in case 
of necessity would find out how to measure every gradation 
of redness or sweetness. 

But the way in which language and natural thought 
operative in language solve the same problem, seems to me 
to be logically very significant. For when, instead of 
attaching a particular name to every single colour of which 
we have actual sensation, we give the privilege of names of 


their own to blue, red, yellow, and a few others, and then 
intercalate the other individual sensations between them as 
bluish red or reddish yellow, this is not merely a shlit for 
approximating to an unattainable exactitude ; rather, as it 
seems to me, it expresses the conviction that only these few 
colours are really fixed points deserving names of their own, 
while the rest must be characterised by approximate ex- 
pressions because they are themselves only approximations 
to these fixed points, or connecting links between them. 
If we really had particular and mutually independent names 
for every single shade of blue, and our ideas answered to 
this form of expression, we should have achieved in a one- 
sided way the separation of each from every other, but we 
should have overlooked completely the positive relations 
which subsist between them all. If on the contrary we 
speak of bright blue, dark blue, black blue, we arrange this 
manifold in a series or a network of series, and in each series 
a third member results from a second by intensification of 
the same sensible change in a common element as that 
which gave rise to the second out of the first. It must be 
already perfectly clear that a presentative activity which did 
not involve this comparison of the diverse, but was confined 
to the bare separation of each from each, would not offer to 
the later operations of thought adequate grounds for con- 
trasting two ideas, as in some way or other cohering, with 
two others as not cohering. We therefore apprehend this 
second act of thought, of which we are here speaking, not 
merely as that of giving simple position to a or , not merely 
as that of simply distinguishing every a from every , but 
also as that of determining the extent and peculiarity of the 
distinction, which is not everywhere the same in degree and 
kind, but is different between b and c and between a and b. 
I do not mean to say that every single idea, a, must be 
accompanied by the developed idea of all its relations to 
the infinite number of all other ideas ; the general accessory 
notion, that every idea is enclosed on all sides in such a 


network of relations, does indeed in our logical conscious- 
nes envelop every idea ; but these relation^ are only 
followed out in each particular case so far as a special 
requirement suggests. 

14. This comparison of the diverse clearly presupposes a 
common element to which in the several members of the 
series specific differences attach. Such a common element 
is usually considered by logic only in the form of a uni- 
versal concept, and in this shape it is a product of more or 
less numerous acts of thought. It is therefore important to 
point out that this first universal, which we find here in- 
volved in the comparison of simple ideas, is of an essen- 
tially different kind ; that it is the expression of an inward 
experience which thought has merely to recognise, and that 
just for this reason it is, as will be seen later, an indispens- 
able presupposition of that other kind of universal which 
we shall meet with in the formation of concepts. We im- 
part the universal concept of an animal or a geometrical 
figure to another person by directing him to execute a 
precisely definable series of intellectual operations, con- 
necting, separating, or relating a number of simple ideas 
assumed to be known ; when this logical work is completed, 
we suppose him to have before his mind the same object- 
matter which we wished to impart to him. But we cannot 
explain by the same means wherein the universal blue or 
the universal colour consists, which accompany our ideas of 
bright and dark blue or of red and yellow. We can indeed 
direct another person to think of all single colours or all 
shades of blue, and by eliminating their differences bring 
out what is common to his ideas in the two cases ; but it is 
only in appearance a logical work which we are here pre- 
scribing ; all that we really call upon him to do is to see for 
himself how he executes the task. How he is to set to 
work to discover whether there really is any common 
element in red and yellow, and how he is to contrive to 
separate it from the differences, this we cannot tell him ; we 


must simply trust to his having an immediate sensation, 
feeling, or experience of the connexion which exists bet^ieen 
red and yellow, of the fact that they contain a common 
element ; his logical work can consist only in the recognition 
and expression of this inward experience. This first uni- 
versal, therefore, is no product of thought, but something 
which thought finds already in existence. 

15. I will insert an observation here which with slight 
modification may be extended to all universals, but is most 
easily illustrated in this simplest instance, the first universal. 
That in which red and yellow agree and which makes them 
both colours cannot be separated from that which makes 
red red and yellow yellow, not separated, that is to say, so 
as to form the content of a third idea similar in kind and 
order to the two compared. It is always, as we know, only 
a single definite shade of colour, only a tone of definite 
height, strength, and quality, which is the object of sensa- 
tion ; and it is only these definite impressions which are so 
repeated in memory as to present substantial and perceptible 
images to consciousness. Universal ideas never have this 
perceptibility. If we try to apprehend the universal element 
of colour or tone, we shall always find that either we have 
before our perception a definite colour and a definite tone, 
only with the accessory notion that every other tone and 
colour has an equal right to serve as a perceptible instance 
of the ever imperceptible universal; or else our memory 
will produce a number of colours and tones in succession, 
with the same accessory notion that it is not these individuals 
that are really meant, but the common element in them 
which cannot as such be apprehended in perception. If 
therefore we understand by idea 1 (as ordinary usage cer- 
tainly inclines us) the consciousness of something standing 
at rest before the mind, or a perception of something 
capable of being presented to it, the universal cannot claim 
to be called an idea. Words like ' colour ' and ' tone ' are 
1 [Vorstellung.] 


in truth only short expressions of logical problems, whose 
solution cannot be compressed into the form pf an idea. 
They are injunctions to our consciousness to present to 
itself and compare the ideas of individual tones and colours, 
but in the act of so comparing them to grasp the common 
element which our sensation testifies them to contain, but 
which cannot by any effort of thought be really detached 
from their differences and made the material of a new and 
equally perceptible idea. 

16. Let us now direct our attention to the differences, 
which, within the first universal, separate the various 
instances of it. It is clear that what distinguishes one 
sensation of warmth from another, a gentler from a louder 
sound, bright from dark blue, is a more or a less of a 
common sensible element, which in itself, undetermined by 
any degree, is no object of perception. We shall find 
ourselves brought back to the same ground of distinction 
in all other ideas ; it is only in giving an account of the 
universal, to which this quantitative comparison applies, 
that we meet with a difficulty, which after the above 
remarks is intelligible. The louder tone is no doubt 
distinguished from the gentler by a certain intensification, 
but so also is the higher from the lower ; yet it is only in 
the former case that we feel able to express directly, by the 
term * strength/ the common element which undergoes Jhis 
change; in the latter we express it by the metaphor of 
height. Red and yellow seem to be still more essentially 
different and underivable one from the other by increase or 
decrease of a common element; only the intermediate 
colours, reddish yellow or yellowish red, are intelligible to 
us as mixtures containing more or less of one or the other. 
Nevertheless no one denies that one of the fundamental 
colours is more nearly related to a second than to a third, 
red to yellow than to green ; and these grades of resem- 
blance cannot be conceived without a more or a less of some 
common element, which we are conscious of in passing 


from one member of the series to the next and from this to 
the third. To determine in each particular case what this 
common element consists in, to decide whether a nurr&er 
of ideas are separated merely by differences in degree of 
one simple universal, or by differences in value of several 
mutually determined ones, and whether accordingly the 
ideas are to be grouped in a linear series or plane-wise or in 
still higher forms, these are all attractive objects of enquiry, 
but they are not objects of logic. For logic it is enough to 
know that some generally applicable and primarily quantita- 
tive determination is the indispensable means for dis- 
tinguishing between the particular instances of a universal. 
And even this determination is something which it is not 
the work of logic to produce, but only to find, recognise, 
and develop. A judgment, a is stronger than bj is indeed, 
as a judgment, a logical piece of work ; but that which 
it expresses, the general fact that differences of degree 
do exist in the same matter, as well as the particular fact 
that the degree of a exceeds that of <, can only be ex- 
perienced, felt, or recognised as part of our inward con- 
sciousness. By whatever artificial contrivances we may 
seek to increase scientifically the exactness of a measurement, 
everything must depend ultimately on the capacity to recog- 
nise two sensuous perceptions as like or as unlike, and not to 
be deceived as to which has the more and which the less. 

17. If inward experience were confined to bringing out 
resemblances and differences in the various object-matters, 
thought would merely be called upon to arrange ideas in an 
unalterable system, like the musical scale, in which all tones 
have once for all their fixed and immoveable places. But 
logic has to do with thought, not as it would be under hypo- 
thetical conditions, but as it is. Now owing to the mechan- 
ism which controls the interaction of its inward states, all 
actual thought has necessarily more opportunities of stimula- 
tion than the above hypothesis would imply ; the manifold 
matter of ideas is brought before us, not only in the 



systematic order of its qualitative relationships, but in the 
rich variety of local and temporal combinations ; and this 
fac^ like the other, belongs to the material wnich serves 
thought in its further operations and must be given it to 
start with. The combinations of heterogeneous ideas pro- 
duced in this way form the problems, in connexion with 
which the efforts of thought to reduce coexistence to co- 
herence will subsequently have to be made. The homo- 
geneous or similar ideas on the other hand give occasion 
to separate, to connect, and to count their repetitions ; and 
to these ideas of unity and multiplicity those of greatness 
and smallness are added where the matter presented is 
continuously extended in space or in time. These three 
pairs of quantitative ideas (for we have already got those of 
more and less) comprise all the standards by which the 
individual instances of any universal are distinguished. 

18. There are two things which I intentionally exclude 
from my consideration. Firstly, all enquiry into the psycho- 
logical character of the growth and development of these 
quantitative ideas in our consciousness, into the order in 
which one of them may condition the origin of another, and 
into the different importance of perceptions of time and 
space in their formation. However attractive these ques- 
tions may be, it would lengthen our way unnecessarily 
to answer them ; logic is not concerned with the manner in 
which the elements utilised by thought come into existence, 
but with their value, when they have somehow or other 
come into existence, for the carrying out of intellectual 
operations. Now this point, which I conceive to have been 
unduly neglected, I wish to emphasize here, and shall 
subsequently keep in view, viz. that all ideas which are to 
be connected by thought must necessarily be accessible 
to one, of the three quantitative determinations which have 
just been mentioned. The other thing which I exclude 
is the investigation of the consequences which may be 
drawn from these quantitative determinations as such : they 


have long ago developed into the vast structure of mathe- 
matics, the complexity of which forbids any attempt to 
re-insert it in universal logic. It is necessary, however, 
to point out expressly that all calculation is a kind of 
thought, that the fundamental concepts and principles 
of mathematics have their systematic place in logic, and 
that we must retain the right at a later period, when 
occasion requires, to return without scruple upon the results 
which mathematics have been achieving, as an indepen- 
dently progressive branch of universal logic. 

10. If we take a general survey of this second act of 
thought, in which I now include that of giving affirmative 
position to the object-matter, that of distinguishing it nega- 
tively from all others, and that of estimating by quantitative 
comparison its differences and resemblances, we may ob- 
serve that the significance of this new logical operation 
is somewhat different from that of the first, by which 
impressions were shaped into ideas. In the former case 
there was a temptation (which, it is true, we resisted) 
to regard the forms of substantivity, adjectivity, and ver- 
bality as modes of apprehension which thought is ready 
to put in practice upon its object-matter before receiving 
any solicitation from it ; but though we set aside this claim 
at once, it remains true that in those forms thought does 
not merely respond to and reproduce the actual current 
of ideas, but gives them the shape without which the logical 
spirit could not accept them. The independence which 
the substantival form gives to its matter, most obviously by 
means of the article, did not itself lie in the fact that 
this matter was a permanent link between changing groups 
of ideas ; nor was the accessory notion of dependence 
expressed by the adjectival form present, as such, in the 
fact which stimulated the mind to characterise it by that 
form ; so that we may continue to assert, in a certain sense, 
that in this first act thought dictates its own laws to its 

D 2 


If, using an expression which we shall otherwise avoid, 
we represent this procedure as a proof of spontaneity, the 
second act of thought has the character of receptivity; it is 
a recognition of facts, and adds no other form to them 
except this recognition of their existence. Thought can 
make no difference where it finds none already in the matter 
of the impressions ; the first universal, as we saw, can only 
be experienced in immediate sensation ; as so experienced 
it can be named, but this is the only contribution which 
logic can make to the further fixing of its character; all 
quantitative determinations, to whatever extent thought 
may develop them by subsequent comparison, always come 
back to an immediate consciousness of certain characteris- 
tics given in the object-matter. I should wish this fact to 
be considered from two points of view. In the first place, 
logic is guilty of a certain carelessness in assuming at almost 
every moment in its later stages the comparability of ideas 
and the possibility of their subordination to a universal, 
without observing that that possibility, and the success of 
its own procedure in general, depends upon this original 
constitution and organisation of the whole world of ideas, a 
constitution which, though not necessary in thought, is all 
the more necessary to make thinking possible. For I must 
repeat that there is no inherent contradiction in supposing 
that every idea was incomparably different from every other; 
that in the absence of all qualitative comparability there was 
no standard of more or less; that the same idea never pre- 
sented itself twice to perception ; and that, as there was no 
repetition of the homogeneous, the ideas of larger and 
smaller also vanished. The fact that this is not the case, 
but that the world of ideas is organised as we have found it 
to be, must be emphasized as of the highest importance ; 
but logic ought not in case of need to appeal to it inci- 
dentally as a self-evident truth derived no one knows 
whence. And this brings me to the other observation 
which I had to make. If thought is a reaction upon a 


stimulus found in the current of ideas, a systematic survey 
of its functions will show clearly at certain points the in- 
fluence exercised upon them by the thinkable world ; as it 
is here the second member in the first triple series of opera- 
tions, so at a later stage also it will be the second member 
of the following more highly developed group in which we 
shall see the peculiar dependence of thought upon the 
material to which it is directed. I do not however claim 
to do more by this preliminary indication than to throw a 
preliminary light over the system which I have followed in 
my exposition ; the system itself can only find its justifica- 
tion in the advantages which in its successive stages it will 
be found to secure. 

C. The Formation of the Concept. 

20. To separate the merely coincident amongst the 
various ideas which are given to us, and to combine the 
coherent afresh by the accessory notion of a ground for 
their coherence, is the further task of thought. It will be 
useful, with a view to making its meaning clear, to rej/iew 
the different senses in which any combination of manifold 
elements occurs in our mental world. In the first place, no 
later intellectual activity is possible, unless the various ideas 
upon which it is to be exercised meet together in one and 
the same consciousness. The fulfilment of this condition is 
secured by the unity of the soul and the mechanism of 
memory, which, by bringing together impressions separated 
in time, makes their interaction possible. This union of 
the manifold may be called the synthesis of apprehension ; 
it is not a logical act; it merely lumps the manifold together 
into a simultaneous possession of consciousness, without 
combining any two of its elements in a different order from 
any other two. Such an order comes in with the second 
form of connexion, the synthesis of perception, that is, with 
figures in space and succession in time, in which the in- 
dividual impressions take up definite and non-equivalent 


positions. This connexion also is supplied by the inward 
mechanism of consciousness without any action pf thought, 
and however firmly defined and finely articulated it may be, 
it exhibits nothing but the fact of an external order, and 
reveals no ground of coherence justifying coexistence in that 
order. From the second stage I pass at once to the fourth, 
to a synthesis in which the last-mentioned requirement 
would be completely satisfied in regard to any given object- 
matter. In such a synthesis we should have before our 
mind, not the mere fact of manifold elements in order, but 
also the value which each element possessed in determining 
the coalescence of the whole. If what we thus apprehended 
were an object in real existence, we should see which were 
the prior, determining, and effective elements in it, in what 
order of dependence and development the others followed 
from them, or what end was to be regarded as their autho- 
ritative centre, involving in itself the simultaneous union or 
successive growth of them all : if, like the figures of geometry, 
it was something which had no reality out of our conscious- 
ness and no growth or development in time, we should here 
too attempt at any rate (though, as we shall see later, with 
limited success) to arrange the elements of the whole in a 
hierarchy in which those that conditioned others should 
take precedence of those that were conditioned, according 
to their stages of dependence. It is easy to see that a 
synthesis of this sort would be neither more nor less than 
the knowledge of the thing ; as the goal of all intellectual 
effort, it lies as far above the province of logic as the first 
and second modes of connexion lay beneath it ; it is in the 
space between that we must place the third and logical form 
of synthesis, the character of which has now to be examined. 
21. When a person who has no special knowledge speaks 
of * credit' or of 'banking,' we trace in these expressions 
his conviction that a number of businesses and institutions 
form a connected whole; but he would not be able to 
say where the nerve of the connexion lies, or what limits 


separate the whole from that which does not belong to it. 
In this accessory notion, that the various elements are not 
merely there in a sort of heap, but form a whole of parts 
with self-imposed limits and a unity included by those 
limits, the general impulse of thought leaves its mark upon 
the given object in a formal way, without as yet attaining 
material fulfilment. If we pass our mental world in review, 
we are in this position as regards a very large part of its 
contents ; indeed we shall be surprised to find that words 
of great significance betray this imperfect apprehension 
of their objects ; for the more complex, important, and 
various any matter is, the more easily will persuasive im- 
pressions derived from repeated observations awaken the 
feeling of its individuality, completeness, and self-inclusive- 
ness, without necessarily giving any real insight into its 
structure. Such words as ' nature/ * life,' ' art/ ( knowledge,' 
' animal,' and many others have no more significance than 
this in ordinary usage; they merely express the opinion 
that a certain quantity, usually not exactly definable, of 
individual objects, attributes, or events, which attach to 
one another, form somehow an inwardly connected whole, 
which can neither have any part taken away without being 
destroyed, nor admit any casual additions within the bounds 
of its unity. But how little the nature of this connexion 
is really known, appears from the failure of the attempt 
to describe the limits which include what belongs to the 
unity and exclude what does not. So long as the logical 
work of holding the manifold together has not gone further 
than this, I should hesitate to speak of * concepts,' though 
I do not attach any value to the invention of a special 
technical term for such imperfect apprehension. Suppose 
we call it an imperfect or growing concept ; then we shall 
not feel that we have' got a perfect or fully developed 
concept, until the vague suggestion of some sort of whole 
has grown into the pervading thought that there is a definite 
ground for the co-existence of these particular attributes, 


in this particular combination and to the exclusion of 
certain others, and that this ground is an adequate one. 

22. The question now arises, how we get at this ground 
and condition. If we merely continued to observe a com- 
posite form a b c d in its isolation, we should never discover, 
however long we looked, which of its parts only coexist, 
which really cohere, and in what degree the existence of 
one depends upon that of another. But if we compare 
abed with other forms like it, that is, with such as we 
are led from it to observe, not by any special logical effort, 
but by the natural current of our ideas, and if we find 
that in abed, a b c f, a b c g, etc., a similar group a b c 
occurs with various dissimilar additions, we regard the 
latter as loose and separable appendages of the permanent 
stem a b c. Nor does the common group a b c contrast 
with the rest merely as the centre to which as a matter 
of fact they attach ; on the general assumption that we 
have before us a whole of interdependent parts, this solid 
kernel becomes the expression of the constant rule which 
allows the accretion of the several accessory elements, and 
determines the manner in which it takes place. If we wish 
for practical purposes to ascertain in any creature, object, 
or arrangement, what is the line which divides what is 
inwardly coherent from casual accessions, we put the whole 
in motion, in the belief that the influence of change will 
show which parts hold firmly together while foreign ad- 
mixtures fall away, and in what general and constant modes 
those parts combine while changing their relative positions 
in particular cases : in this sum of constant elements we 
find the inner and essential cohesion of the whole, and 
we expect it to determine the possibility and the manner 
of variable accretions. The first of these methods, that 
of bringing out the common element in different instances 
when at rest, is the one which has been usually followed 
by logic, and has led to the formation of the logical 
universal ; I should give the preference to the other, that 


of determining the element which maintains itself in the 
same instance under changed conditions; for it is only 
the assumption that the group a b c, the common element 
in several groups of ideas, will also be found thus to 
maintain itself, which strictly justifies us in regarding these 
coexisting elements as coherent, and as the ground for the 
admissibility or inadmissibility of fresh elements. 

23. Abstraction is the name given to the method by 
which the universal is found, that method being, we are 
told, to leave out what is different in the particular instances 
compared and to add together that which they possess in 
common. If we look at the actual procedure of thought, 
we do not find this account confirmed. Gold, silver, 
copper, and lead differ in colour, brilliancy, weight, and 
density ; but their universal, which we call metal, is not 
found upon comparison by simply leaving out these differ- 
ences without compensation. Clearly it is no sufficient 
definition of metal to say negatively, it is neither red nor 
yellow nor white nor grey; the affirmation, that it has at 
any rate some colour, is equally indispensable ; it has not 
indeed this or that specific weight, this or that degree of 
brilliancy, but the idea of it would either cease to have 
any meaning at all, or would certainly not be the idea 
of metal, if it contained no thought whatever of weight, 
brilliancy, and hardness. Assuredly we do not get the 
universal image of animal by comparison, if we leave out 
of our minds entirely the facts of reproduction, self-move- 
ment, and respiration, on the ground that some animals 
produce their young alive, others lay eggs, others multiply 
by division, that some again breathe through lungs, others 
through gills, others through the skin, and that lastly many 
move on legs, others fly, while some are incapable of any 
locomotion. On the contrary, the most essential thing 
of all, that which makes every animal an animal, is that 
it has some mode or other of reproduction, of motion, and 
of respiration. In all these cases, then, the universal is 


produced, not by simply leaving out the different marks 
p l and / 2 , q 1 and ^ 2 , which occur in the individuals com- 
pared, but by substituting for those left out the universal 
marks P and <2, of which /* / 2 and q^ q 1 are particular 
kinds. The simple process of leaving out only takes place 
when one of two individuals compared actually possesses 
no species of a mark P, of which some species is a neces- 
sary mark of the other. Thus we suppose, whether rightly 
or wrongly does not matter, that we cannot find in plants 
any trace of sensation and self-movement, both of which 
are essential to all animals; we do therefore form the 
universal idea of organic being from a comparison of plant 
and animal by leaving out these marks without compen- 
sation. If we went thoroughly into the facts, we should 
perhaps find occasion, not indeed in this instance but in 
many similar ones, to continue to ascribe two marks jointly 
to both the objects compared, but to assume them to be at 
zero in the plant, while in the animal they always occur in 
an appreciable quantity. To express the matter somewhat 
differently, it may be asserted from the point of view of 
logic that compensation by the corresponding universal for 
omission of individual marks is the regular rule of ab- 
straction, while the uncompensated omission applies to 
exceptional cases, where we can find no logically common 
mark, of which the presence and absence of some individual 
mark might be held to constitute different species. So 
formulated, our rule of abstraction covers these cases of 
mere omission; on the other hand, a rule which made 
omission its sole starting-point could find no way to bring 
in compensation afterwards ; and the importance of com- 
pensation in forming the universal will be confirmed at 
every step in the later stages of logic. 

24. After the considerations urged in the preceding 
section, the necessity of which to what was to follow will 
now be clear, the apparent circle involved in the injunction 
to form universals by putting together universals, will not 


give serious offence. We have seen that the universal 
marks P and Q which we require here, the ' first universal ' 
of the section referred to, come to us without logical effort 
as simple facts of observation in our mental life ; and just 
for this reason they can be applied in building up this 
second universal, which we do produce by logical effort. 
That the yellow of gold, the red of copper, and the white 
of silver are only variations of a common element which 
we proceed to call colour, this is a matter of immediate 
sensation; but to a person who could not be made 
sensible of it, it could never be explained by logic either 
that these particular impressions are species of this uni- 
versal, or what is meant by a universal as such and the 
relation of its particular to it. It is just this point to 
which I would again draw attention here, that the im- 
mediate perception of a first universal and the application 
of some kind of quantitative ideas is the condition of the 
formation of the second universal in all cases, not only in 
those like metal where there is no difficulty in regarding 
the marks of colour, brilliancy, and hardness as stable pro- 
perties of that which they describe, but also where, as in 
the case of the animal powers of reproduction and motion, 
they are merely short adjectival descriptions of conditions 
which we cannot think completely but by means of mani- 
fold relations between various related points. It is easy 
to convince oneself by an analysis (which I only leave the 
observant reader to make for himself because it threatens 
to be a long one) that all differences between animals, even 
in these respects, issue ultimately in quantitative deter- 
minations, whether of the force with which some identical 
or similar process takes place in them, or of the number of 
related points between which it takes place, or of the 
variations in form to which it is liable owing to variations 
in the number of these related points, the intimacy of their 
relations, and their relative positions in space and time, 
these last, like the rest, being measurable variations. If 


we take away this quantitative gradation and comparability, 
which extends, though of course in different ways, to every- 
thing, whether simple properties, or their relations, or com- 
binations of events simultaneous or successive, the formation 
of a universal by comparison of different groups of ideas, 
would, at least in the sense in which it has any value for 
thought, be impossible. 

25. I will now mention some traditional technical ex- 
pressions. If we provisionally give the general name of 
concept (notio, conceptus) to the composite idea which we 
think as a connected whole, the sum of individual ideas or 
marks (notae) a, b, <r, d, etc., through which a concept S is 
fully thought and distinguished from all other concepts 2, is 
called its ' content ' (mater la) ; while its * extent ' (ambitus, 
sphaerd) is the number of individual concepts j 1 , J 2 , s"*, etc., 
in each of which the content of S, that is, the group of 
marks #, , c, d, in some one or other of their possible 
modifications, is contained. The colour, , weight, <, 
elasticity, d, and the like, would together form the content 
of metal, S, while copper, /, silver, j 2 , gold, s\ and the like, 
taken together, form its extent. It is usual also to speak 
of the individual marks #, , c, as ' coordinated ' in the 
content of S, and of the individual species s\ s\ /, as ' co- 
ordinated ' in the extent of S : the relation of the species 
s\ s 2 , 5 3 , to the universal itself which forms their genrs, is 
called * subordination, 7 while both the species and the 
genus are said to be ' subsumed ' under each of the 
universally expressed marks, which make up the content 
of S, and consequently also of s\ j 2 , s*. Lastly, it is asserted 
that the extent and content of every concept vary inversely ; 
the greater the content, that is, the number of marks which 
the concept imposes upon all its subordinate species, the 
smaller is the number of species which fulfil this require- 
ment; the smaller the content of -S 1 , the greater is the 
quantity of individuals possessing the few marks necessary 
to make them species of S or bring them within its extent. 


If therefore we compare the universal concept S with a 
similar universal T> and look for a third universal U to 
which both of them belong as species, and if we continue 
this process, the higher each universal concept /^stands in 
the scale, that is, the farther it is removed from the 
concepts S and T originally compared, the poorer will it be 
in content and the larger in extent ; and conversely, if we 
descend from the highest universals /^through Fand 7, 
S and T, to the species of S and lower, the content will 
increase with the decreasing extent and become greatest in 
those completely individual ideas to which logic hesitates 
to give the name of concept at all. 

26. The value of these distinctions is unequal, but on 
the whole slight. I will begin what I have to say about 
them by fixing the terminology which I shall myself use in 
future. I speak of any composite matter s as conceived or 
as a concept, when it is accompanied by the thought of a 
universal S, which contains the condition and ground of 
the coexistence of all its marks and of the form of their 
connexion. After this explanation we shall not hesitate to 
speak of concepts of perfectly individual things (singular 
concepts, in the old logical terminology), and we believe 
this to be quite consistent with the usage of language. For 
when we observe a new object s for the first time, and, not 
content with the perfectly clear sensible perception of it, go 
on to ask what it really is, we clearly want to know the rule 
which connects the perceived marks in the observed fact 
and converts them into a coherent whole of a definite and 
predictable character. If we then find that this s is S, an 
animal or a plant, we suppose ourselves to have a con- 
ception of s ; it is the idea of it which is raised into a 
concept by the accompanying thought of the universal S. 
Every proper name is an illustration of this. 'Alcibiades,' 
for human thought, never means merely a multiplicity of 
differently coloured points, which are combined in space in 
a definite though not quite invariable outline, and resist the 


attempt to separate them ; nor does the name express 
merely the accessory notion that this multiplicity in some 
unexplained way forms a whole ; it suggests to the mind a 
definite general image of a man or a human being, which 
lays down the lines for our view of the connexion of the 
observed marks with one another and with the future 
behaviour to be expected from them. A view so deter- 
mined cannot be appropriately called either a perception, 
or an idea merely, but only a singular concept. 

27. On the other hand, it seerns to me quite out of place 
to call the universal S itself, the accompanying thought 
of which makes the individual into a concept, without any 
reservation a universal concept, S may have the form of a 
concept, but by no means always has it ; often it remains a 
mere general image, the thought of which is indeed accom- 
panied by the thought of its connected wholeness, but does 
not exhibit the organic rule of the connexion. The name 
' man ' as ordinarily used expresses no more than an image 
of this kind ; reflexion, by subordinating it to the universal 
( animal,' easily makes it into a concept ; but then * animal ' 
remains a general image, which only the naturalist, for the 
uses of his science, converts into a concept by thinking 
{ organic being ' along with it. It is upon such incomplete 
logical activity, which brings into relief only a single link in 
the chain, the connexion of the individual with its nearest 
universal, but leaves all beyond it in darkness, that the 
concepts which occur in ordinary thinking are based; as 
however scientific investigations, to which logic is primarily 
intended as an introduction, do really aim at extending the 
conceptual form from the concept itself to the higher 
universals under which it successively falls, it is enough to 
have made the above remark without rigidly enforcing it, 
and I shall follow ordinary usage in conceding the name 
of concept to those general images as well. I can do this 
the more easily because the name ( concept ' does not seem 
to deserve in logic that exalted significance which the 

Chap. I.] COORDINA TION. 4 7 

school of Hegel has given it, and in which it claims to 
express the knowledge of the essential nature of the object. 
The difference between logical forms and metaphysical ideas 
must be taken into account here as elsewhere. There may 
be a privileged concept, which follows the thing itself in its 
being and development, or takes up a point of view at the 
very centre of the thing, the fountain-head of its self-deter- 
mination and self-organisation ; but it is not the function of 
logic to reserve its concept-form for so very select a filling. 
By the logical concept we understand such a form of 
apprehending any matter of thought, from whatever point 
of view, that consequences admit of being drawn from it 
which coincide again at certain points with results flowing 
from that matter, that is, from the thing itself ; and as the 
thing projects itself differently at every different point of 
view, there may be various equally right and equally fruitful 
logical concepts of the same object. We may therefore 
continue to call * concept ' any apprehension which, though 
only with the help of a general image which is not further 
analysed, has the effect of bringing the given object under 
a rule of behaviour which agrees, when applied, with its 
actual behaviour. 

28. The asserted coordination of marks in the content 
of the concept raises serious difficulties. To begin with, 
it is a misfortune that we have no appropriate name for the 
elements of which we compose the concept ; for ' mark J 
and ' part ' only apply in certain cases. They give rise to 
the current delusion that the elements of a concept are 
universally of equal value, connected in the same way each 
with the whole and each with each. The ordinary instances 
of logic, taken from simple natural objects, are specially 
calculated to lead us into this error. It is true that gold is 
yellow only in the light, ductile only under a certain power 
of traction, heavy only for the body upon which it presses ; 
but these various modes of behaviour easily present them- 
selves to our imagination as stable properties, collected in a 


definite point of space, and inhering, in a manner identical 
but otherwise unexplainable, in the reality which on their 
account we call gold. Here the name * marks' is appro- 
priate, and here the marks are certainly coordinated in the 
content as has been asserted ; but this coordination merely 
means that they are all equally indispensable to the whole, 
but have not any other sort of order. If we leave such 
simple instances, and consider concepts like 'triangle,' 
* animal,' or 'motion,' we require, in order to think them 
properly, a quantity of part-ideas which are no longer 
mutually equivalent, but have to be placed in the most 
various relations to one another. The three sides of a 
triangle are not merely there as well as the three angles ; 
they must form the angles by their intersections : the concept 
of motion does not merely contain the part-ideas of place, 
change, direction, and speed ; direction and speed are, each 
in a different sense, determinations of change ; place, being 
that which is left behind, can least of all be called a mark 
of the concept ; it is a point of reference for the idea of 
change, to which its relation is expressed by that of the 
genitive to the nominative which governs it. To follow 
out these points in detail would take too long, but it would 
evidently lead us to the conviction that, as a rule, the marks 
of a concept are not coordinated as all of equal value, but 
that they stand to each other in the most various relative 
positions, offer to each other different points of attachment, 
and so mutually determine each other; and that an appropri- 
ate symbol for the structure of a concept is not the equation 
Sz=a + & + c+d, etc., but such an expression as S=J? 
(a, b, r, etc.), indicating merely that, in order to give the 
value of S, a, , c, etc., must be combined in a manner 
precisely definable in each particular case, but extremely 
variable when taken generally. If in any particular 

instance S= a[t> G sin d~\ + (e }V~h, this formula, how- 

ever foolish it would be if it professed to mean anything 


more, would give a better picture than the above inadequate 
formula of addition of the different ways in which the several 
marks a, b, t, etc. contribute to the construction of S as a 

29. No objection need be made to the coordination of 
s 1 , s 2 , /*, copper, gold, silver, within the sphere of S, metal ; 
on the other hand, attention should be drawn to the great 
difference of value between the subordination of the species 
to the genus, and that of the universal along with its 
species to the universal marks a, b, (ductile, coloured, etc.). 
The nature of the universal, metal, completely dominates 
the nature of its species, gold and copper, and no property 
of the latter escapes its influence ; many things are yellow 
or red, but the glistening red and yellow of copper and gold 
belong to metal alone ; many things are ductile, but the 
amount and other peculiarities of the ductility exhibited by 
gold and copper are heard of only in metals ; and only 
metallity explains their degree of specific gravity. Similarly 
the universal animal determines every property and every 
movement of its species; animals move, grow, and rest 
differently from plants and lifeless things. If we symbolise 
the universal metal by a circle , the smaller circle of gold, 
s\ lies entirely within it, and by the side of this, separate 
from it but also completely inside S, the circles j- 2 , j 3 , copper 
and silver. Applying differently two names which are 
generally used as equivalents, I describe the true subordina- 
tion to a dominant universal as subordination to the genus, 
while I call the subordination of gold to yellow or ductile 
subsumption under the mark. These universal marks obvi- 
ously do not rule and penetrate the whole nature of gold ; 
each of them expresses only one side of it, which it shares 
with other objects of an entirely different kind, from which, 
so far as logic can see, no sort of inferences can be drawn 
as to the other properties of gold. Thus the lesser circle s, 
gold, occurs only in a particular place in the larger G t 
yellow, and intersects it without lying wholly within it ; G 

LOGIC, Vou I. E 


is similarly intersected in other places by the circles of other 
yellow objects, and they all remain partially outside it. 

30. Starting from the universal S, which was the rule for 
s\ .r 2 , j- 3 , the original objects of comparison, we were able to 
mount to higher and higher universals T, U, V, W. In 
natural history, where such a series is of value, its several 
members in an ascending scale have been named species, 
genus, family, order, class : there is however a difference of 
opinion as to what functions a universal concept must per- 
form in order to represent even a species or a genus, and 
the other names are applied still more divergently, and 
always from points of view depending for their justification 
on the special nature of the subject-matter. If we dispense 
with this plea, the plea from the side of the specialist for the 
significance and importance of these distinctions, the only 
way to give some sort of fixed logical value to species and 
genus is as follows. The only thing which suggests to the 
natural mind to look for a universal, is the comparison of 
individual instances which are not identical but similar. 
To seek for a concept which included under it cucumbers 
and mathematical principles, could only be an ingenious 
joke ; but all varieties of human beings, big and little, old 
and young, fat and thin, black and white, provoke the 
natural mind to the search. Their sensible appearances 
produce similar images, at the corresponding points of which 
only such marks occur as are immediately felt to be species 
of the same universal mark, such as hardness or colour; 
and the relations between any two of these points are in all 
cases merely modifications, differing in degree and amount, 
of one and the same universal relation. The comparison of 
individual men, therefore, produces a universal image ; not 
indeed in the sense that the universal man can really be 
painted, but in the sense of the illustrations in a natural 
history, which purport by one camel or horse to exhibit all 
camels or horses clearly to perception, in a form which is 
more than a mere scheme or symbol ; or again in the sense 


of geometry, in which a drawn triangle, though necessarily 
individual with others existing beside it, yet represents all 
these others, and in a similarly perceptible form. But this 
possibility vanishes when we ascend to higher universals, in 
which these universal images are themselves included in 
their turn as species : the universal mammal, which is 
neither horse nor camel nor is otherwise named, cannot 
even be drawn in a schematic form, any more than the 
polygon can which has neither three, four, or any other 
definite number of sides. Thus these higher universals are 
no longer apprehended in perception, but only in thought, 
by means of a formula or equation, which prescribes 
essentially the same relation between various related points, 
but leads to quite different perceptible configurations, 
accordingly as the previously undetermined values of these 
points and their various connexions are differently deter- 
mined in thought. I would then call a universal which 
still admits of an image, a species, and the first of those 
which can only be expressed by a formula, a genus, in 
agreement, as I believe, with the instinct of language, and 
incidentally also with the old terminology of Aristotle ; for 
in his choice of the words fldos and yei/os he was no doubt 
determined by their original meanings ; flSos, the species, 
which includes only individuals under it, is the common 
element in the look or appearance of things, while yeW 
comprehends things which differ in form, but in their pro- 
cess of growth, or, if they have no growth in time, in the 
regulative connexion of their parts, obey the same law and 

31. It remains to consider the last of the assertions 
mentioned above, that of the inverse ratio between the con- 
tent and extent of concepts ; this seems to me to be untrue 
where its truth would be important, and to be comparatively 
unimportant where it is true. The number of marks, of 
which we compose our concepts, is not infinite ; the words 
of language, numerous but not innumerable, suffice to de- 

E 2 


note them. It may therefore easily happen that a group of 
them, say ikl, occurs in several universal concepts, 5 jTand 
F", at once, without its therefore representing a higher uni- 
versal containing all species of S jTand V. We may class 
cherries and flesh under the group i k I of red, juicy, edible 
bodies, but we shall not suppose ourselves thereby to have 
arrived at a generic concept of which they deserve to be 
called species. I do not say that in giving exclusive pro- 
minence to such groups there is always as little sense as in 
this absurd instance; we shall see later how valuable the 
process may be ; it helps to show, what is often useful and 
necessary, that different subjects, though otherwise quite 
foreign to one another and not subsumable under any 
common generic concept, are nevertheless, in consequence 
of a single or a few common marks, jointly liable to certain 
inevitable consequences. If then anyone chooses to go on 
to call these groups of marks universal concepts, he is cer- 
tainly right about the inverse ratio of their content and 
extent : the fewer members there are in the group, the more 
sure will it be to occur in all sorts of concepts, and again, 
the greater the number of different ideas compared, the 
smaller will be the group of marks in which they all agree. 
Of the true universal, on the other hand, which contains the 
rule for the entire formation of its species, it may rather be 
said that its content is always precisely as rich, the sum of 
its marks precisely as great, as that of its species themselves; 
only that the universal concept, the genus, contains a num- 
ber of marks in a merely indefinite and even universal 
form ; these are represented in the species by definite 
values or particular characterisations, and finally in the 
singular concept all indefiniteness vanishes, and each uni- 
versal mark of the genus is replaced by one fully determined 
in quantity, individuality, and relation to others. It is true 
that instances may be alleged against the universal validity 
of this assertion, like that mentioned above of organic being, 
to the concept of which we subordinate plants and animals ; 


it may be called a logical caprice to retain the marks of sen- 
sibility and motivity in this concept, with the tacit reservation 
that they are both at zero in plants. But what this instance 
properly shows is rather, that the higher universals, from the 
genus upwards, really cease to be true universal concepts, and 
pass over into groups of conditions, imposing uniform conse- 
quences upon various genera, more properly so called. 
The concept of organic being is such a group of marks, 
ikl) which does not occur in any independent form of its 
own, but in the genera in which it does occur, plants and 
animals, gives rise necessarily to the same results. 

32. By the preceding remarks I neither hope nor aspire 
to bring about a permanent change in the traditional ter- 
minology : they were intended merely as helps to a clearer 
insight into the structure of concepts in general. With the 
same object I add the following. I express the genus G, 
so far as its concept gives the rule of combination for a 
number of individual marks A B C, etc., by F [A B C\ 
and I assume that each of the marks admits of particular 
forms, which we may call a 1 a* a* . . P & fr . . <r l <rV; also 
that the principle of combination F has freedom to assume 
various forms, of which we may indicate three by/J (/>, and 
f. Now as the marks ABC may be of very different value 
for the whole G, it is possible that the different values 
assunied e.g. by A may be of decisive importance for the 
configuration of the whole, and may also exercise a trans- 
forming influence upon the combination of the other marks. 
The consequence of this may be that, as A assumes one or 
other of its values, the organisation of the whole, F, changes 
from one of its particular modes to another ; the sum total 
of the species of G would then be, 

G=f(a l B C. ..) + < (<?B C. . . ) -f f (c?B C\ 
omitting for shortness' sake to express the corresponding 
changes in B and C. These decisive marks, a 1 a 1 a*, are in 
this case the specific differences, differentiae specificae. Thus 
Aristotle, who gives them the name of Sia$o/>u, when he sub- 


ordinates man to the genus animal, usually describes the 
faculty of rational thought as that peculiar characteristic, a 1 , 
of the universal psychical life, A, by which man is dis- 
tinguished from all other animals ; to this we may now add, 
following out what I have indicated above, that this d> not 
only separates man from brutes, but also determines the 
values of B and C peculiar to him, as also the mode of 
their combination, i.e. the general character by which man 
is distinguished from the brutes with their peculiar organisa- 
tion $ or f. It may further happen that the particular 
values which one or more of the generic marks, have as- 
sumed in a single species, are possible in this and no other 
species, and that yet they have no important influence upon 
the shaping of its other marks, and do not therefore repre- 
sent the nature of it in all its aspects. Such a mark is called 
by Aristotle property, i8w; it is what we call a characteristic; 
Aristotle gives risibility as a property of man, Hegel, in a 
similar sense, the ear-lap ; both distinguished man from the 
brutes, but without exhausting his nature. There are also, 
according to Aristotle, marks which do riot belong to the 
rigid constitution of a concept, but indicate something 
which comes in contact with or happens to it ; every verb 
which says, e.g. Socrates ' is sitting' or ' standing,' is an 
example. Translators torment themselves in vain to find 
an equivalent for both the real and the etymological sense 
of Aristotle's expression crvpfie&rjKos ; what is important and 
true in it answers completely to what we call state; that 
this word does not nevertheless cover the usage of Aristotle 
seems to me to be the fault of an inexactitude of his own, 
which it is scarcely worth while to enter into. As to the 
relation in fact between the concept as a whole and this 
species of mark, its consideration belongs to the theory of 
the judgment. In the introduction of Porphyrius to the 
Aristotelian logic there is material enough for further 
reflexion, though indeed of a mostly unprofitable sort, 
about the likenesses and differences of the logical determi- 


nations here touched upon ; we have used them primarily 
to illustrate the complex organisation of concepts, and with 
this view have not always agreed with Aristotle in the form 
of our exposition. 

33. And now, where do we et to at last if we go on 
looking for higher and higher concepts above those which 
we have already found ? What form does the entire system 
of our concepts assume if we suppose this task completed ? 
It must be a structure resting on a broad base, formed by 
all singular concepts or ideas, and growing gradually 
narrower f as it rises. The ordinary view, in fact, gives it 
the form of a pyramid, ending in a single apex, the all- 
embracing concept of the thinkable. I cannot see much 
point in this notion ; it rests entirely upon that unmeaning 
subsumption under a mark, the logical value of which we 
have already depreciated. A single step suffices to bring 
everything at once under the head of the thinkable; we 
may spare ourselves the trouble of climbing up to this 
result by a pyramidal ladder ; and moreover the result itself 
ignores in the most absolute and unmeaning way every- 
thing which gives substance and character to thought. If 
on the other hand we follow the method of subordination 
to the genus, and arrange the manifold only under such 
universals as still imply the notion of universally regulating 
its specific conformations, we arrive not at one but at 
several ultimate concepts not reducible to one another, in 
which we are not surprised to recognise those very meanings 
of the parts of speech which at the outset we found to be 
the primary logical elements. All substantives go back to 
the radical concept of something, all adjectives to that of 
quality, verbs to that of becoming, and the rest to that of 
relation. It is true that all these radical concepts have the 
common mark of being thinkable ; but there is no common 
genus over them of which their several essences form 
species, nor does any one of them occupy this position in 
regard to the rest ; it is not possible to apprehend some- 


thing as a species of becoming, or becoming as a species of 
something. From this point of view the entire structure 
of our concepts rises like a mountain-chain, beginning 
in a broad base and ending in several sharply defined 

Transition to the form of the Judgment. 

34. It was this image of a conceptual world building 
itself up without a break, upon which the vision of Plato 
dwelt. The first to recognise the eternal self-identity of 
every concept and its significance as against the variableness 
of the real world, he might well feel the charm of tracing 
out all the simple elements of thought, of combining all 
that could be combined, and of setting up in the organic 
whole of a world of ideas the eternal pattern of which the 
created world is an imperfect imitation, But neither he 
nor his successors have attempted actually to execute this 
essentially impossible task : still less should we now be 
inclined to regard its execution as desirable. And this 
not only because reality, things as they are, suggests riddles 
too many and too hard to leave us any time for drawing up 
an inventory of what might be but is not; for even a 
perfect knowledge of the ideal world would give us little 
support in understanding the real. The utmost that we 
could attain by such means would be merely the image of a 
fixed order, in which simple and composite concepts stood 
side by side, each unchangeably self-identical and each bound 
to its place in the system by invariable relations to all the 
rest ; whereas what reality shows us is a changing medley 
of the most manifold relations and connexions between 
the matter of ideas, taking first one form and then another 
without regard to their place in the system. This great 
fact of change does not cease to be a fact because, in the 
spirit of antiquity, we find fault with it as an imperfection 
compared with the solemn rest of the world of ideas : the 


current of our thoughts is perpetually bringing it before us 
again, and the mind, receiving as it does from that current 
the stimulus to activity, has to exert itself to reduce even 
these changeable coincidences to principles of coherence. 
The next advance of logic is determined by this fact. 

35. There are different considerations which lead us to 
take the same step next. When new marks, of which 
we were not before conscious in a concept, attach them- 
selves to it without its apparently being changed, we are 
directly stimulated to ask what ground can be conceived for 
such a variable connexion of the two. But also when we 
compare different instances of a universal, in the universal 
marks of which we have already included the possibility of 
many particular ones, it may still be asked on what ground 
a particular mark in each instance coheres with the rest of 
the content, and why this particular mark is privileged 
above all the others which remain absent, though, as species 
of the same universal, they might equally well be present. 
Lastly, as we think of every concept as uniting a number of 
marks, and these marks, though not essentially related 
as members of one and the same systematic series, but 
rather heterogeneous and foreign to one another, neverthe- 
less determine each other and in their combination influ- 
ence the accession of others, the question again recurs, 
whflt is the ground of the apparent coherence in this co- 
existence of heterogeneous elements. We are conscious 
that when, in considering the concept, we attributed to 
a certain combination of marks this position of a dominant 
logical substance, operating in a number of different or 
changing forms, we required and presupposed a view which 
we have yet to show to be logically practicable. This then 
is our present problem, either to break up these presup- 
posed combinations again, or, if they can be justified, 
to reconstitute them, but in a form which at the same time 
expresses the ground of coherence in the matter combined. 
In seeking to solve this problem, the form in which thought 


will move will obviously be that of the judgment In this 
a permanent conditioning member, the whole Content of 
a concept, appears as subject over against the variable or 
conditioned members or the sum of them, as predicates ; 
the relation of the two, explaining and justifying their con- 
nexion, lies in the copula, that is, in the accessory notion 
which, more or less fully expressed in language, holds 
together the two members of the sentence. 


The Theory of the Judgment. 

Preliminary observations on the meaning and customary 
division of Judgments. 

IN accordance with the general plan of my exposition, 
I should now have to develop the various forms of judg- 
ment systematically as members of a series of intellectual 
operations, each one of which leaves a part of its problem 
unmastered and thereby gives rise to the next. Before 
beginning this attempt, I must say a few words about other 
usual modes of treatment, and my reasons for deviating 
from them. 

36. Every judgment formed in the natural exercise of 
thought is intended to express a relation between the 
matters of two ideas, not a relation of the two ideas them- 
selves. Of 'course some sort of relation between the ideas 
follows inevitably from the objective relation in the matter 
which they represent ; but it is not this indispensable 
relation in the mental media through which we endeavour 
to grasp the matter of fact, but this matter of fact itself, 
which is the essential meaning of the act of judgment 
When we say, * gold is yellow/ it is indisputable that in this 
judgment our idea of gold lies within the sphere of the idea 
of yellow, and that accordingly the predicate is of wider 
extent than the subject; but it was certainly not this that 
we intended to express by the judgment. We wanted 
to say that yellow itself belongs as a property to gold itself, 
and only because this relation of fact is already presupposed 


to exist (whatever difficulties this may involve), can it be 
reproduced in a sentence in which the idea qf gold is 
contained by that of yellow. Logic indeed has already 
drawn attention to the fact that we are not quite right even 
in making this sentence ; appealing from what we express 
to what we mean, it teaches that the subject also from 
its side limits the too extensive predicate; gold is not 
yellow simply, but golden yellow, the rose rosy red, and 
this particular rose only this particular rosy red. But even 
with this correction the imperfection of this whole view 
of the judgment is not mended ; for it does not , tell us 
what is after all the relation between the two members so 
corrected, and it loses sight entirely of the great possible 
variety in the modes of their connexion. Thus gold is not 
yellow in the dark ; its colour therefore only attaches to it 
under a condition, that of the presence of light ; and if we 
wished to connect this new experience with the previous 
one in the phraseology of the view which we are now 
considering, we should have to say, the idea of gold lies 
simultaneously within the spheres of that which is yellow in 
the light and of that which is not yellow in the dark ; but 
this form of expression seems to me only to betray a 
disposition to leave the really important point, the mention 
of the conditional relation, and to go off upon results which 
are true but quite without significance. Doubtless these 
relations of extension between the ideas combined in the 
judgment have their logical value ; but where the want 
of them is felt, they are not so difficult but that they can be 
mastered at the moment without special effort : to give 
them a chief place in the consideration of the judgment 
seems to me to be as erroneous as it is wearisome. 

37. The technical expressions of logic 'point to the view 
which I have taken here. In the judgment above the subject 
in the sentence, that is, the grammatical subject, is the 
word gold, the subject in the judgment, the logical subject, 
is, not the idea of gold, but gold ; for it is to this only that 


yellow belongs as that which is predicated of it, and pre- 
dicated in a definite sense indicated by the copula. On the 
other hand, the idea of yellow is not a property of the idea 
of gold in the same sense in which yellow is of gold ; the 
one idea is not affirmed or predicated of the other; the 
relation which exists between them is primarily no more 
than this, that whenever, or whenever under certain con- 
ditions, the one idea, gold, is found, there the other idea, 
yellow, is also found, but that the former is not always 
present when the latter is. But to explain and express 
what it is which makes this relation possible, justifiable, or 
necessary, is the problem of the logical judgment alone, 
and it solves the problem by exhibiting through its copula 
the relation between the object-matters of the two ideas, a 
relation due to that which the ideas represent and differing 
in different cases. On the other hand, it is only between 
these object-matters that a logical copula is conceivable ; 
between the ideas there is no relation but that of the 
psychological connexion mentioned above, and that of the 
monotonous, unmeaning inclusion of the one within the 

38. It is now clear that for us there can be only so many 
essentially different forms of judgment as there are essentially 
different meanings of the copula, that is, different accessory 
notions which we form of the connexion of the subject with 
its predicate, and to which we give more or less complete 
expression in the syntactical form of the sentence. Thus 
many other distinctions which meet us in logic have no use 
or place in our systematic survey, though they may still 
have a logical value of some other kind. To' secure clear- 
ness in what is to follow, therefore, it is desirable to give a 
preliminary explanation of traditional views ; but I think I 
may confine it to that division of judgments to which Kant 
has given currency in Germany, though it is itself of much 
older date. According to Kant, as we know, the character 
of every judgment is determined in four respects, quantity,, 


quality, relation, and modality, and in each respect every 
judgment has necessarily one of three mutually exclusive 
forms. I may exclude the third member of this division 
from these preliminary considerations, for relation (between 
subject and predicate), in respect of which Kant distinguishes 
categorical, hypothetical, and disjunctive judgments, clearly 
concerns just those essential characteristics of the judgment 
which we are looking for, and which I shall have sub- 
sequently to expound myself. If the categorical judgment 
connects its subject *S and its predicate P absolutely ', as the 
phrase is, or on the simple model of the relation oa thing 
to its property, while the hypothetical assigns P to S, not 
immediately, but only on the assumption that a certain 
condition is fulfilled, and the disjunctive gives S no definite 
predicate, but imposes on it the necessity of choosing 
between several mutually exclusive ones, there is no doubt 
that in each of these three forms the sense of the copula, 
the mode of connexion between S and P, is different and 
peculiar ; these three will form the series of judgments 
which we shall have subsequently to construct ; only the 
nine remaining ones call for the following preliminary 

39. In respect of their quantity judgments must be either 
universal or particular or singular. If we express these 
distinctions by the usual formulae, ' all S are P,' * som^ S 
are Pf c this S is /V it is clear that they indicate merely 
the different extents to which a connexion between S and P 
is supposed to hold good ; the nature of the connexion in 
all the cases is the same, and must be the same, because 
the universal judgment, according to this view of its 
meaning, admits of being formed by summing the singular 
and particular ones, and must therefore be perfectly homo- 
geneous with them. Thus the quantitative description 
applies to the subject only, and has no reference to the 
logical relation between it and its predicate ; it is therefore 
of importance where the connexion of ideas requires the 


application of a judgment, the import of which depends 
upon the circuit over which it holds good ; but no special 
advance in logical activity is indicated by these distinctions 
as they are here formulated. I say ' as they are here 
formulated,' because certainly the quantitative differences 
of judgments are really connected with important logical 
differences in the mode of connexion between S and P\ 
for doubtless that which belongs to all S has also a different 
hold upon the nature of its subject from that which belongs 
only to some; but the quantitative formulation of the 
judgment, which merely counts the subjects, just fails to 
seize this important accessory notion, and makes the relation 
of the predicate to its subject, often in violation of the fact, 
appear the same in all cases. 

4O. In respect of quality Kant distinguished affirmative, 
negative, and limitative judgments. Nothing is clearer than 
that the two sentences '*$ is PJ 'S is not P,' so long as they 
are supposed to be logically opposed to one another, must 
express precisely the same connexion between S and P, only 
that the truth of that connexion is affirmed by the one and 
denied by the other. It is useful, though certainly not 
necessary, to make this clear to ourselves by splitting each 
of these judgments into two. We think of a certain relation, 
whatever it may be, between S and P expressed in the judg- 
ment < is P* as an idea still open to question ; this relation 
forms the object-matter upon which two opposite judgments 
are passed ; the affirmative gives it the predicate of validity 
or reality, the negative refuses it. In the connexion of our 
thoughts it is of course of the greatest importance "which of 
these judgments is subsequently passed upon 'a given con- 
nexion between and P\ but this difference does not give 
rise to two essentially different kinds of judgment as such ; 
validity or invalidity are rather to be considered, in regard 
to the question before us, as predicates of fact to which the 
whole content of the judgment forms the subject. This 
content itself can be expressed in a form as yet neither 


affirmative nor negative in the interrogative sentence, and 
this indeed would take the third place amongst the three 
qualities of judgment more appropriately than the limitative 
or infinite judgment, which is supposed to attribute a nega- 
tive predicate to the subject by a positive copula, and is 
usually expressed in the formula c S is not-/*.' Much acu- 
men has been expended even in recent times in vindicating 
this form of judgment, but I can only see in it an unmean- 
ing product of pedantic ingenuity. Aristotle himself saw 
clearly enough that such expressions as i not-man ' are no 
concepts they are not even apprehensible idea". The 
truth is that, if ' not-man ' means all that it ought logically 
to mean, that is, everything that is not man, triangle, melan- 
choly, sulphuric acid, as well as brute and angel, it is an 
utterly impossible feat to hold together this chaotic mass of 
the most different things in any one idea, such as could be 
applied as a predicate to a subject. Every attempt to affirm 
this unthinkable not-P of -5* will be found by an unsophisti- 
cated mind to end in denying the thinkable P of the same 
Sj instead of saying, ' spirit is not-matter,' we all say, c spirit 
is not matter.' Even in cases where in natural thinking we 
seem really to make a limitative judgment, as e.g. when we 
say ' doctors are non-combatants/ we are in truth making 
only a negative one. For this not-P has not here the 
meaning which the limitative sentence would give it; ac- 
cording to that, horses, wagons, triangles, and letters would 
be non-combatants; what is meant is only human beings 
who belong to the army but are declared to take no part in 
fighting. Thus there is never any necessity to the natural 
mind for forming limitative judgments ; every inference 
which could be drawn from '5 is not-P' can also be drawn 
from '* is not P.' It is not worth while to spend more 
words on this point ; obvious vagaries in science must not 
be propagated even by a too elaborate polemic. 

41. Through the forms of modality different values are 
supposed to be given to the relation which is conceived 


to hold between S and P\ the problematic judgment ex- 
presses it aj merely possible, the assertorial as real, the 
apodeictic as necessary. But these new properties are treated 
quite independently of the way in which judgments have 
been already determined from the other three points of 
view. After it has been fixed whether a given judgment 
J connects its elements in categorical, hypothetical, or dis- 
junctive form, after it has been decided nj whether it affirms 
or denies the relation conceived in one of those forms, and 
after the extent of the subject to which the predicate applies 
has been limited by the expression of quantity, it is still 
held to be an open question whether the judgment so com- 
posed will be problematic, assertorial, or apodeictic. To 
treat the matter thus is to confess openly that the possibility, 
reality, or necessity, spoken of here, stand in no connexion 
with the logical construction of the judgment. All these 
judgments, which are usually expressed in the formulae 
' S may be /V ' S is P,' ' S must be P,' are entirely the 
same as regards the validity which they give to their con- 
tents by logical means ; they are all merely assertions of 
the person who makes them, and are distinguished only by 
their object-matter. This, the possibility, reality, or ne- 
cessity of a relation between S and P, they express either 
without any grounds at all, or upon grounds derived from 
right reflexion upon the facts, which they do not then allow 
to appear in any way in their logical structure ; just for this 
reason they need additional auxiliary verbs, in order to 
express independently what does not lie in the form of the 
judgment itself. In more developed connexions of thought 
such judgments of course have their value; for what is 
wanted is often to compress results of previous reflexion 
into the shape of simple assertions, without perpetually 
repeating the grounds upon which they rest; here these 
auxiliary verbs are in place, expressing in the form of a now 
familiar fact the possibility, reality, and necessity which 
once had a logical justification. But for the separation of 



essential forms of judgment and their systematic arrange- 
ment, the only modality that could be of vakie would be 
one which, instead of going its own way independently of 
the logical nexus of the other judgments, grew out of that 
nexus itself, and expressed the claim to possible, real, or 
necessary validity, which the content of the judgment de- 
rives from the mode in which its elements are combined. 

42. It would be useless to ask for such a modality, if we 
could not show the possibility of it. I will therefore an- 
ticipate somewhat what I have to say later. The proposition, 
'all men must die/ is usually held to be apodeictic ; I con- 
sider it merely assertorial \ for it states only, and does not 
give grounds for, the necessity of which it speaks ; so far as 
its form goes it does not even decide whether all men die 
for the same reason, or everyone for a special reason, so 
that the various conditions agree merely in the fact that 
they leave no one alive. And yet what we had meant 
by the sentence was, not only that all men as a matter 
of fact die, but that the extension of mortality to all has 
its ground in the universal concept of man, in the nature of 
humanity and this thought we do in fact express by the 
general form of the judgment 'man dies'; for the sense 
of this judgment, the difference of which from the ordinary 
universal I shall come back to, is not of course that the 
universal concept man dies, but that everything dice which 
is included under it, and for the reason that it is so in- 
cluded. Every hypothetical judgment, again, gives in its 
protasis the ground for what is stated in its apodosis, 
and is therefore in my sense an apodeictic form of judg- 
ment j the apodosis here is not simply asserted, but asserted 
conditionally upon the validity of the protasis ; but, pre- 
supposing that validity, the content of the apodosis is 
no longer a mere fact, but a necessity, with the same right 
with which every consequence necessarily follows from its 
conditions. Similar remarks might be made, if they would 
not be too long for this preliminary section, about the 


disjunctive judgment; and thus we should have found in 
the three forms of relation three forms also of apodeictic 

43. I will guard myself against a misunderstanding, though 
it would be so gross that I am almost ashamed to do so. 
The form which we give to the content of a judgment 
can never guarantee its truth to fact ; this always depends 
upon whether the relations between the elements of the 
content itself are truly such as the form of the judgment, in 
order to ascribe to them a certain sort of validity, has 
to presuppose. This holds good of the ordinary modality 
no less than of that which we would put in its place. In the 
ordinary form of the apodeictic judgment, ' S must be /*,' 
any nonsense may be expressed without thereby becoming 
sense ; and it is equally open to us to misuse the judgments 
which I call formally apodeictic, and say l man is omni- 
potent/ ' if it rains everything is dry, 7 ' every triangle is 
either curved or sweet or hasty-tempered/ These latter 
forms of judgment, then, do not, any more than the former, 
make every connexion of concepts which is put into them 
true or necessary; the significance of them lies merely 
in showing the formal conditions under which we may 
ascribe demonstrative certainty to a given content, if that 
content is in itself such as to satisfy them. And here our 
view of modality differs to its advantage from the ordinary 
one. The latter merely tells us that there is demonstrative 
knowledge, and that, if we have got it, we can express it in 
the form ' S must be P'\ but it does not tell us how know- 
ledge must look, and what its internal structure ,must be, in 
order to be demonstrative and to justify this expression. 
Our plan on the other hand does show us this ; we find 
that there are three forms of relation between *$ and P, 
which, when they, exist, lead to necessary knowledge ; 
endeavour to bring your ideas into one of these forms ; 
either frame general judgments and look for the P which is 
already implied in the conception of a genus S ; this P then 

F 2 


belongs necessarily to every species of S: or form hypo- 
thetical judgments, and show that the addition to of 
a condition X gives rise to a P which would not otherwise 
be present ; then this P holds necessarily of every S which 
comes under the same operation of the same conditions : 
or lastly form disjunctive judgments ; as soon as you have 
brought a question to a definite ' either . . or/ the thing 
is settled, and all that is now wanted is experience to deter- 
mine, in each particular instance, which of two predicates, 
P or (?, will be true and necessarily true. There are no 
other ways of arriving at necessary knowledge, and every 
judgment which we express in the form, 1 S must be PJ 
remains merely an assertion, the matter of which, if it is 
convincing, has always been originally apprehended in one 
of those three ways. 

44. Thus far I have spoken only of apodeictic judg- 
ments : the ambiguity of the ordinary theory of modality is 
still more striking in the case of problematic judgments. 
The proposition, 'all bodies can be set in motion by 
adequate forces/ may have any one of the three modalities 
ascribed to it with about equal right. Firstly, as a state- 
ment which does not add the grounds upon which it is 
made, it is assertorial : but what it states is not a real 
occurrence, but the possibility of an unreal or only con- 
ceived one, and this is enough according to traditional 
usage to give it the name of problematic : lastly, it may be 
called apodeictic, because it ascribes a property to all 
bodies, a property therefore which can be wanting in none 
and is accordingly necessary to each : in fact, this judgment 
contains the reality of the necessity of a possibility. From 
which point of view are we to choose its name ? I should 
be in favour of regarding it as an assertorial judgment, 
reckoning the necessary possibility as part of the matter 
asserted. As however the same view may be extended to 
all problematic judgments of the ordinary form, the ques- 
tion arises whether there is any form of judgment at all 


which, as such, deserves to be called problematic. Inter- 
rogations and prayers have been alleged as instances, for 
neither of them really asserts anything ; the connexion of *S 
and P which forms their content seems to be presented to 
the mind as no more than a floating possibility. I doubt 
however whether they can be considered as specific logical 
forms at all. For ultimately interrogation must be dis- 
tinguished from prayer, and the distinction can only lie in 
the fact that the conscious attitude of the questioner to his 
question is different from that of the petitioner to his peti- 
tion. Suppose the import of the question to be, * I do not 
know whether S is P* and that of the petition, ' I wish that 
S were P" ; it would of course be very pedantic to say that 
the speaker himself must always analyse what he says into 
this bipartite form, but still, if we take his consciousness as 
a whole, it must contain in both cases two different states, 
tempers, dispositions, or whatever we call them, which, if 
we wished to express them, could only be expressed in 
those ways. If this is so, it is clear that both judgments 
contain a principal sentence of an assertorial form, which 
says nothing about the content of the judgment but merely 
indicates the attitude of the speaker to it ; the other and de- 
pendent sentence, introduced by the conjunctions ' whether ' 
or ' that,' comprises the whole content, without saying any- 
thing about the nature and degree of its validity. It is for 
this reason that I do not consider the dependent sentence 
either to be a problematic judgment ; for it is not enough 
that the account of the nature of the import should be 
merely absent ; the import ought to be explicitly confined 
to mere possibility. As to the prayer, it might further be 
said that it contains the possibility of what is prayed for 
and nothing else, whereas the question, as it may be a 
question about possibility itself, does not always do even 
that : in both moreover the assumption of the possibility of 
a conceived connexion between S and P could only be 
reckoned as a state of the speaker's mind, and would not 


lie in the logical form of the judgment. I should rather 
consider this dependent sentence to express ^without any 
modality the mere content of a judgment ; and it is just 
because no complete judgment can be expressed without 
claiming possibility, reality, or necessity for its import, that 
these sentences void of modality never occur indepen- 
dently, but are always governed by some other independent 
sentence which asserts one of those modalities of its con- 

45. According to our view those judgments only could 
be called problematic which by their logical forpn charac- 
terise a conceived relation between S and P as possible 
and only as possible. This is done by all quantitatively 
particular and singular judgments. All that is directly ex- 
pressed by sentences of the form, * Some are /V ' Some 
S may or must be P,' l This S is P' or ' may or must be 
P' is the actual, possible, or necessary occurrence of P in 
certain cases of S -, they leave it doubtful how the matter 
stands with the other cases of S which are not mentioned ; 
for S as such, therefore, it is only the possibility of each of 
these three relations to P which is expressed, and these 
particular sentences are equivalent to the assertions, ' <S may 
be P possibly/ * S may be P] * S may be P necessarily.' I 
therefore call particular sentences problematic in respect of 
the universal *$"; the fact that they are clearly also assertorial 
in respect of the some S of which each speaks, does not at 
all militate against my view ; it only shows us that in fact 
the only way of recognising a certain relation between -5* 
and P to be merely possible is by observing that the rela- 
tion does, may, or must hold good of some S and not of 
others. There are therefore certainly no independent prob- 
lematic judgments, which are not assertorial in respect of a 
part of their universal subject in so far as they affirm of it a 
possible, actual, or necessary predicate. 

46. Lastly, it is easy to see that, on the one hand, the 
1 may' and * must' of the ordinary problematic and apodeictic 


judgments and the 'is' of the assertorial by no means 
suffice to express all material differences of importance in 
the truth of their several -contents, and that on the other 
hand, just for this reason, they lump together very different 
relations under the same expression. Firstly, what modality 
have such sentences as these, * S will be P,' ' S ought to 
be jP,' ' S may be P,' ' S has been P' ? No one of them 
affirms reality, but the unreal which is past in the last is 
something quite different from that which is permitted, 
enjoined, or future, in the others : in the third it is possible, 
in the second its possibility is doubtful, in the first its reality 
is inevitable, while in the last it is at once irretrievable and 
unreal. If all these shades of meaning had been taken into 
account, the forms of modality might have been correspond- 
ingly increased in number. On the other hand, how entirely 
different in meaning are the similarly formed sentences, 
'It can rain to-day V 'The parrot can talk/ 'Every quad- 
rangle can be divided into two triangles.' In the first case 
we have a supposition which is possible because we know 
no reason to the contrary ; next a capacity which exists 
upon conditions which need not have existed ; lastly a 
necessary result of an operation which we may carry out or 
not as we please. I will not multiply these instances, as 
might be done indefinitely ; to attempt to analyse them all 
would be as foolish as to undertake to work out beforehand 
all possible examples in a mathematical text-book. In 
practice, indeed, it is just from these material varieties of 
meaning in the expressions in question that our inferences 
are drawn but we have no resource except to observe in 
each particular instance what we have before, us ; whether 
it is a possibility which may be tentatively assumed in the 
absence of proof to the contrary, or a well-grounded 
capacity resting securely upon its conditions ; whether it is 

1 [' Es kann hente regnen ; der Papagei kann reden ' ; in English we 
say, ' It may rain to-day,' so that the difference of meaning is represented 
by some difference of form.] 


a necessity due to the presence of imperative reasons, or 
one arising from a command, a purpose, a duty, or lastly 
one of those combinations of possibility, reality, and ne- 
cessity which we touched upon above. 

The series of the forms of Judgment. 

A. The Impersonal Judgment, The Categorical Judgment, 
The Principle of Identity. 

47. There can be no doubt that in the series of the forms 
of judgment the categorical comes before the hypothetical 
and the disjunctive. We could have no- occasion for 
making the occurrence of a predicate P in a subject S 
dependent on a previously fulfilled condition, unless we 
had already had experiences of the presence of P in some 
S and its absence in others. Equally little can we think of 
prescribing to S the necessary choice between different 
predicates, until previous experiences have established the 
constant relation of S to a more universal predicate, of 
which the proposed alternatives are specific forms ; and 
these experiences too would find their natural expression in 
a judgment of the form ' S is P.' The structure, moreover, 
of the hypothetical and disjunctive judgments exhibits per- 
manent traces of this dependence : however complex they 
may be in particular cases, the general scheme to which 
they are reducible is always that of two judgments of the 
form ' S is PJ combined, either as protasis and apodosis or 
as mutually conclusive members, so as to form a single 
complete assertion. But the question may be raised 
whether a still simpler form must not precede the categorical 
judgment itself in the systematic series. The sentence 
'AS* is P' cannot be uttered until the current of ideas has 
informed us of an 5 with a fixed position and recognisable 
character of its own, to which a P can be added in thought 
as a predicate. Now this will not always be the case ; 


indeed it may be questioned whether the discovery of the 
definite 5, which is to serve as subject to a categorical 
judgment, does not always presuppose experiences of S in 
a less developed form, and their translation into logical 
equivalents. This question, which relates to the psycho- 
logical growth of thought, I leave unanswered here ; for our 
present purpose the fact is enough that even our fully 
developed thought has preserved a form of judgment which 
performs this simplest of functions, that of giving logical 
setting to a matter of perception without regarding it as a 
modification or determination of an already fixed subject. 
This is the impersonal judgment, which, as the first act of 
judging, I here treat as a preliminary stage to the cate- 

48. I do not think it necessary to defend at length the 
logical import of the impersonal judgment against the 
opinion which would make it merely the linguistic expres- 
sion of perception itself, without involving any logical 
activity. The natural sound which a man who is shivering 
with cold makes when he cowers against another, is a mere 
sign of this sort, which only serves to give tongue to his 
feeling ; but as soon as he expresses his discomfort in the 
sentence * it is cold,' he has undoubtedly performed an act 
of thought. By giving to the content of his perception, 
which in itself is undivided, this bipartite form of a predi- 
cate related to a subject by a copula, he expresses that he 
can think of it as a perceived reality in no other form 
than this. It is true that he is not in a position to give the 
subject an independent content; he only indicates its 
empty place and the fact that it requires filling, either by 
the indefinite pronoun, or in other languages by the third 
person of the verb, which he uses instead of the infinitive : 
it is true also that the whole content of the perception 
which he expresses falls into the predicate alone : and it is 
true, lastly, that the copula which he puts between them 
has not as yet the sense of a definitely expressible relation ; 


it only keeps formally apart what is substantially inseparable 
and interfused. But it is just by this attempt to bring 
about an articulation to which the matter of perception will 
not yet lend itself, that the impersonal judgment expresses 
all the more clearly the instinct of thought, that everything 
which is to be matter of perception must be conceived as a 
predicate of a known or unknown subject. 

49. I will now explain why I have here spoken repeat- 
edly of perception 1 . The indefiniteness of the subject in 
the impersonal judgment has been interpreted to mean 
that it merely expresses in substantival form what is 
expressed in verbal form by the predicate. I do not 
doubt that anyone who is asked what he means by 'it,' 
when he says ' it rains/ or l it thunders,' can easily be 
driven to say, ' the rain rains,' or ' the thunder thunders.' 
But I believe that in that case his embarrassment makes 
him say something different from what he really intended 
by his impersonal judgment. It seems to me to lie in 
the essence of such a judgment that he really looks upon 
the determinate matter in question as attaching to an in- 
determinate subject, the extent of which is much wider 
than that of the predicate; and if he uses several such 
expressions one after another, * it lightens,' ' it rains,' * it is 
cold,' though he does not expressly intend to say that the 
indefinite pronoun means the same in all those cases, he 
would certainly, if he understood himself correctly, give this 
answer rather than the former one. This 'it' is in fact 
thought of as the common subject, to which the various 
phenomena attach as predicates or from which they pro- 
ceed , it indicates the all-embracing thought of reality, 
which takes now one shape, now another. This has been 
rightly felt by those who found in the impersonal judgment 
a judgment of existence, and transformed the sentence 'it 
lightens' into 'the lightning is.' It is only the transforma- 
tion itself which seems to me unnatural ; we never express 
1 [' Wahrnehmung.'] 


ourselves in this way ; the unsophisticated mind does not 
think of the phenomenon as if it were already something 
before it existed, of which we could speak, and of which 
among other things we could assert reality; on the contrary, 
it regards the particular reality in question as a phenomenon, 
a predicate, a consequence, proceeding along with others 
from an antecedent and permanent, though quite inexpres- 
sible, subject. Though however we cannot accept this 
explanation, it is so far right as that every genuine imper- 
sonal judgment expresses an actually present perception, 
and is therefore as regards its form an assertorial judgment. 
Such genuine judgments are to be distinguished from other 
modes of expression which begin with the indefinite 'it 7 as 
subject, but immediately fix its content by an explanatory 
sentence, as, e. g. ' it is well that this or that should be done.' 

60. The more definitely the mind emphasizes the neces- 
sity of the subject to which the predicate is to attach, the 
less can it rest content with an expression in which this 
demand is unsatisfied. It is not part of my logical task, as 
I have already said, to describe the processes of comparison 
and observation by which our ideas of those subjects are 
gradually formed, which we require to take the places of 
the indefinite 'it' in the various impersonal judgments; 
I have only to point out the logical form in which this 
requirement is satisfied. Most of the simple instances with 
which logic usually begins its illustration of the judgment 
in general, are in the familiar form of the categorical judg- 
ment ' S is /y e.g. 'gold is heavy,' 'the tree is green/ 
4 the day is windy.' No explanation is needed as regards 
this form ; its structure is perfectly transparent 1 and simple : 
all that we have to show is, that this apparent clearness 
conceals a complete enigma, and that the obscurity in 
which the sense of the copula in the categorical judgment 
is involved will form a motive that will carry us a long way 
in our successive modifications of logical activity. 

61. A certain embarrassment is at once observable as 


soon as we ask in what sense 6* and P are connected in the 
categorical as distinct from the hypothetical an d, disjunctive 
judgments. A common answer is, that the categorical 
judgment asserts of P absolutely, but this answer is only 
negatively satisfactory, i. e. so far as it denies of the cate- 
gorical sentence the idea of a condition and the idea of an 
opposition between mutually exclusive predicates ; but when 
we know what this form of judgment does not do, the state- 
ment that it joins P to S absolutely gives us no positive 
information as to what it does do.- Such a statement in 
fact merely expresses the greater simplicity of the cate- 
gorical copula as compared with that of the hypothetical 
and disjunctive judgments; but this simpler connexion 
must still have a determinate and expressible sense of its 
own, distinguishing it from other conceivable forms of 
connexion equally simple or more complicated. The ne- 
cessity of explaining this sense appears most simply from 
the fact, that, of all connexions of S and P, the complete 
identity of the two would be that which most obviously 
deserved the name of absolute. Yet it is just this which 
as a rule is not intended in the categorical judgment : 
'gold is heavy' does not mean that gold and weight are 
identical; equally little do such sentences as 'the tree 
is green,' 'the sky is blue,' identify the tree with green 
and the sky with blue. On the contrary, we are at oains 
to express our real meaning in such judgments by saying, 
' P is not S itself, but only a predicate of ,' or * S is 
not P^ it only has P? We thus admit that we are thinking 
of a definite and distinguishable relationship between S 
and P, and it only remains to make really clear what 
constitutes this 'having' which we oppose to 'being/ or, 
in more logical language, wherein we have to look for 
that relation of a subject to its predicate which we wish to 
distinguish from the relation of identity. 

52. Plato was the first to touch this problem; his 
doctrine, that things owe their properties to participation 


in the eternal universal concepts of those properties, was 
rather an' inadequate answer to a metaphysical question 
about the structure of reality, than an explanation of what 
we have in our mind when we establish a logical relation 
between subject and predicate. Aristotle made the right 
treatment of the question possible by observing that the 
attributes are primarily enunciated of their subjects; this 
at any rate established the fact that it is a logical operation 
of the mind which refers the matter of the one concept 
to that of the other ; but more than this name of enun- 
ciation, "arqyopeu/, from which that of the l categorical' 
judgment and that of the Latin equivalent * predicate' 
are derived, even Aristotle did not discover. He escaped 
indeed a confusion of later logic; he did not reduce 
the connexion which he supposed between S and P from 
a logical operation to a mere psychical occurrence, thus 
making the relation between the two consist only in the 
fact that the idea of P is associated in our consciousness 
with that of S: for him the sense of the judgment and 
the ground for making it was a real relation between the 
matters of the two ideas. But he did not tell us how 
precisely S is affected by the fact that we enunciate P 
of it; he made the enunciation itself, which can really 
do nothing but recognise and express this real relation, 
stand for the very relation which it had to recognise. 
Now it is easy to see that this fusion is quite inadmissible ; 
it is impossible merely to enunciate the concept * slave' of 
Socrates in such a way that the enunciation itself should 
settle the relationship in which the two concepts stand 
to one another: what we really mean by a judgment is 
always, that Socrates is or is not a slave, has or has not 
slaves, liberates or does not liberate slaves. It is one or 
other of these possible relations which constitutes what 
is enunciated in each case, and it is only a matter of 
linguistic usage that, when we speak of enunciating the 
latter concept of the former, we choose tacitly to under- 


stand only the first relation, viz. that Socrates is a slave. 
The relation, therefore, of 6" to P in a categorical judgment 
is not distinguished from other relations by saying that 
P is enunciated of ; the truth rather is that the meaning 
of this enunciation, in itself manifold, is determined by 
the tacit supposition that P is enunciated of S as a predicate 
of a subject. It still remains a further question, what con- 
stitutes this peculiar relation. 

53. We moderns are accustomed on this point to hold 
to the doctrine of Kant, who represented the relation of 
a thing to its property, or of substance to its* accident, 
as the model upon which the mind connects S and P in 
the categorical judgment. This statement may have a 
good meaning in the connexion in which Kant made it, 
but it does not seem to be available for the logical question 
before us. I will not here raise the point whether the 
idea of the relation between substance and attribute is 
itself so clear and intelligible as to dissipate all obscurity 
from the categorical judgment; it is enough to remind 
ourselves that logical judgments do not speak only of 
what is real, of things ; many of them have for their subject 
a mere matter of thought, something unreal, or even im- 
possible. The relation existing between the real thing 
as such and its properties obviously cannot be transferred 
in its full sense to the relation of subjects to their pre- 
dicates, but only in the metaphorical or, as we may say, 
symbolical sense. To speak more exactly, the only common 
element in these two kinds of relation is the formal one, 
that in both the one of the related members, thing, or 
subject, is apprehended as independent, the other, property 
or predicate, as dependent upon the former in the way 
of attachment or inherence. But in regard to the thing, 
metaphysic has at any rate exerted itself to show how 
there can be properties which are not the thing and yet 
attach to it, and what we are to suppose this attachment 
to consist in; whereas in regard to the relation between 


subject and predicate we find no corresponding account 
of the sense in which the one inheres in the other. The 
appeal to the relation between thing and property, there- 
fore, does not help logic at all ; the question repeats itself, 
How much of this metaphysical relation survives as a 
logical relation expressible in the categorical judgment, 
if the thing be replaced by something which is not a thing, 
and the property by something which is not a property ? 

54. Without adding any more to these customary but 
unsuccessful attempts to justify the categorical judgment, 
I will state the conclusion to which we are driven : this 
absolute connexion of two concepts S and P 9 in which the 
one is unconditionally the other and yet both stand over 
against each other as different, is a relation quite imprac- 
ticable in thought ; by means of this copula, the simple ' is ' 
of the categorical judgment, two different contents cannot 
be connected at all; they must either fall entirely within 
one another, or they must remain entirely separate, and the 
impossible judgment S is P ' resolves itself into the three 
others, 'S is S,' 'P is P,' ( S is not P.' We must not stumble 
too much at the startling character of this assertion. Our 
minds are so constantly making categorical judgments of 
the form ' S is PJ that no doubt what we mean by them 
will eventually justify itself, and we shall soon see how this 
is possible. But the categorical judgment requires such a 
justification ; taken just as it stands it is a contradictory and 
self-destructive form of expression, in which the mind either 
represents as solved a hitherto unsolved problem, the 
determination of the relation between S and P, or so 
abbreviates the discovered solution that their connexion is 
no longer visible. On the other hand we are met by the 
consciousness that all our thought is subject to a limitation 
or has to conform to a law ; by the conviction that in the 
categorical judgment each constituent can only be conceived 
as self-same. This primary law of thought, the principle of 
identity, we express positively in the formula A~A, while 


in the negative formula, A does not = non-J, it appears 
as the principle of contradiction to every attempt to make 

56. I will not interrupt my exposition here by remarks 
which would have to be repeated later upon the various 
interpretations which this first law of thought has received ; 
I will confine myself to stating exactly what sense I shall 
myself attribute to it in opposition to many of those inter- 
pretations. In the case of an ultimate principle, which 
limits the whole of our thinking, it is obvious that with the 
application of thought to different groups of objects it will 
be transformed into a number of special principles, which 
exhibit its general import in the particular forms in which it 
applies to the particular characteristics of those groups and 
has an important bearing upon them. The consequences 
thus drawn from the principle of identity, some of which 
are quite unexceptionable while others are by no means so, 
must be distinguished from the original sense of the principle 
itself, and do not belong to this part of logic. Thus it is 
quite useless to expand the expression of the law into the 
formula, Everything can have at the same moment and in 
the same part of its whole self only one predicate A, and 
cannot have at the same time a predicate non-A contrary or 
contradictory to A, This statement is certainly correct, but 
it is no more than a particular application of the principle 
to subjects which have the reality of things, are composed 
of partsj and are capable of change in time. On the other 
hand it is incorrect to distinguish, as is often done tacitly 
and not less often explicitly in formulating this principle, 
between consistent predicates, which can belong at the same 
time to the same subject, and others which cannot because 
they are inconsistent with one another and with the nature 
of the subject. In the applications of thought, of course, 
this distinction too has its validity, when it has justified 
itself before the law of identity ; but, taken as it stands, 
that law knows nothing of predicates which, though different 


from S, are still so far consistent with it that they could be 
combined with it in a categorical judgment ; on the con- 
trary, every predicate P which differs in any way whatever 
from *$*, however friendly to *$* it might otherwise be con- 
ceived to be, is entirely irreconcileable with it ; every judg- 
ment of the form, ' S is P] is impossible, and in the strictest 
sense we cannot get further than saying, *S is S' and 
' P is P? The same interpretation of the principle must 
also be maintained against other metaphysical inferences 
which are drawn from it. It may be that in the course of 
metaphysical enquiry it becomes necessary to make such 
assertions as, What is contradictory cannot be real, What is 
must be unchangeable, and the like : but the logical law of 
identity says only, What is contradictory is contradictory, 
What is is, What is changeable is changeable : all such 
judgments as make one of these concepts the predicate of 
another require a further special explanation. 

B. The Particular Judgment. The Hypothetical Judgment. 
The Principle of sufficient Reason. 

56. It would be wearisome to stay longer at a point of 
view in which we could never permanently rest : we will 
follow thought to the new forms in which it tries to bring 
its categorical judgments into harmony with the law of 
identity. Judgments of the form 'S is P y are called syn- 
thetical, when P is understood to be a mark not already 
contained in that group of marks which enables us to 
conceive S distinctly; they are called analytical when P, 
though not identical with the whole of 6", yet belongs 
essentially to those marks the union of which is necessary 
to make the concept of S complete. In the analytical 
judgment people have found no difficulty; but the syn- 
thetical attracted attention at an early period, and Kant's 
treatment of it in particular has recently made it con- 
spicuous. He too however was mainly interested in ac- 
counting for the possibility of synthetical judgments a priori^ 



i.e. such as assert an existing and necessary connexion 
between S and a concept P not indispensable to S, without 
the need of appealing to the experience of its actual 
occurrence : as to synthetic judgments a posteriori, which 
merely state that such a connexion between two not 
mutually indispensable concepts is found or has been 
found in experience, he regarded them as simple expressions 
of facts and therefore free from difficulty. These distinctions 
may be fully justified within the circle of enquiry in which 
Kant moved ; but our logical question as to the possibility 
of categorical judgments extends to all three forms with 
equal urgency. The necessity of justification before the 
principle of identity is only more obvious in the case of 
the a priori synthetical judgment, which formally con- 
tradicts that principle \ but it holds good of the a posteriori 
also. For a judgment does not simply reproduce the fact 
like a mirror; it always introduces into the observed 
elements of the fact the thought of an inner relation, 
which is not included in the observation. Experience 
shows us only that S and P are together \ but that they 
are inwardly connected, as we imply when we predicate P 
of S in the judgment, is only the interpretation which our 
mind puts upon the fact. How this relation can subsist 
between subject and predicate in general, and between 
S and P in particular, is just as obscure after experience 
has shown the coexistence to be a fact as when we assert it 
in anticipation of experience. Lastly, analytical judgments 
raise the same difficulty. However much yellow may be 
already contained in the concept of gold, the judgment 
1 gold is yellow' does not assert merely that the idea of 
yellow lies in the idea of gold, but ascribes yellowness to 
gold as its property; gold must therefore have a determinate 
relation to it, which is not the relation of identity. This 
relation has to be explained, and the question still remains, 
What right have we to assign to S a P which is not S, as 
a predicate in a categorical judgment ? 


57. The only answer can be, that we have no right : the 
numberless categorical judgments of this form which we 
make in daily life can only be justified by showing that 
they mean something quite different from what they say, 
and that, if we emphasize what they mean, they are in fact 
identical judgments in the full sense required by the prin- 
ciple of identity. The first form in which we get a hint of 
this in the natural course of thought is that of quantitative 
judgments in general, which I shall in future call shortly 
particular, and consider as the first form of this second 
group of judgments. Under this title I include not only 
the traditional forms, such as, ' all S are /y ' some 8 are 
/y this 'S is PJ which have for their subject a number of 
instances of the general concept 6', but those also which in 
various other ways limit to definite cases, and therefore 
particularise, the universal application of the connexion 
between S and P, whether by particles of time (now, often, 
etc.), or by those of space (here, there, etc.), or again by 
a past or future tense of the verb, or lastly by any kind of 
accessory idea, imperfectly expressed or not expressed at all. 
In the general formula of the categorical judgment, ' S is 
/y it looks as if the universal *S" were the subject, the 
universal P its predicate, and the constant, unchangeable, 
and unlimited connexion of S and P the import of the 
whole judgment. If on the other hand we supply explicitly 
what is suggested, or at any rate is meant, by these par- 
ticularising accessory ideas, we find that the true subject is 
not the universal S, but S, a determinate instance of it; 
that the true predicate is not the universal P y but n, a 
particular modification of it; and lastly that the relation 
asserted is not between S and P, but between 2 and n, 
and that this relation, if the supplementary ideas are correct, 
is no longer a synthetical, nor even an analytical one, but 
simply one of identity. A few instances will make this 

58. We say, * some men are black/ and suppose ourselves 

G 2 


to be making a synthetical judgment, because blackness is 
not contained in the concept of man. But the* true subject 
of this sentence is not the universal concept * man ' (for it is 
not that which is black), but certain individual men ; these 
individuals, however, though they are expressed as merely 
an indefinite portion of the whole of humanity, are yet by 
no means understood to be such an indefinite portion ; for 
it is not left to our choice what individuals we will take out 
of the whole mass of men ; our selection, which makes them 
* some' men, does not make them black if they are not so 
without it; we have, then, to choose those men, and we 
mean all along only those men, who are black, in short, 
negroes ; these are the true subject of the judgment. That 
the predicate is not meant in its universality, that on the 
contrary only the particular black is meant which is found 
on human bodies, is at once clear, and I shall follow out 
this remark later ; here I will only observe that it is merely 
want of inflexion in the German expression which deceives 
us as to its proper sense ; the Latin ' nonnulli homines sunt 
nigri' shows at once by number and gender that * homines' 
has to be supplied to ' nigri.' The full sense, then, of the 
judgment is, ' some men, by whom however we are only to 
understand black men, are black men '; as regards its matter 
it is perfectly identical, and as regards its form it is only syn- 
thetical because one and the same subject is expressed from 
two different points of view, as black men in the predicate, as 
a fragment of all men in the subject. Again, we say, * the 
dog drinks.' But the universal dog does not drink ; only a 
single definite dog, or many, or all single dogs, are the 
subject of the sentence. In the predicate too we mean 
something different from what we express : we do not think 
of the dog as a sort of ever-running syphon ; he does not 
drink simply, always, and unceasingly, but now and then. 
And this 'now and then' also, though expressed as an 
indefinite number of moments, is not so meant; the dog 
drinks only at definite moments, when he 'is thirsty or at 


any rate inclined, when he finds something to drink, when 
nobody stops him \ in short, the dog which we mean in this 
judgment is really only the drinking dog, and the same 
drinking dog is also the predicate. Again, { Caesar crossed 
the Rubicon '; but not the Caesar who lay in the cradle, or 
was asleep, or was undecided what to do, but the Caesar 
who came out of Gaul, who was awake, conscious of the 
situation, and had made up his mind ; in a word, the 
Caesar whom the subject of this judgment means is that 
Caesar only whom the predicate characterises, the Caesar 
who is crossing the Rubicon, and in no previous moment of 
his life was he the subject to whom this predicate could 
have been attached. It is obvious moreover to every 
capacity that when he had crossed the river he could not go 
on crossing it, but was across, so that in no subsequent 
moment of his life either can he be the subject intended in 
this judgment. I will give two more examples, which Kant 
has made famous. It is said that the judgment, * a straight 
line is the shortest way between two points,' is synthetical, 
for neither in the concept 'straight' nor in that of t line' is 
there any suggestion of longitudinal measure. But the 
actual geometrical judgment does not say of a straight line 
in general that it is this shortest way, but only of that one 
which is included between those two points. Now this fact, 
the fact that its extension is bounded by two points, (and it 
is only with this qualification that it forms the true subject 
of the sentence) is the ground, in this case certainly the 
satisfactory ground, for assigning the predicate to it. It is 
easy to see that the concept of a straight line a I between 
the points a and b is perfectly identical with the concept of 
the distance of the two points ; for we cannot give any 
other idea of what we mean by ' distance in space^ than 
this, that it is the length of the straight line* between a and 
b. There is not therefore a shorter and a longer distance 
between a and , but only the one distance a <, which is 
always the same. On the other hand, we can speak of 


shorter and longer ways between a and b\ the concept of 
way implies merely any sort of progression whidi leads from 
a to b ; as this requires the getting over of the difference 
which separates b from #, there can be no way leading from 
a to b which leaves any part of this difference not got over ; 
accordingly, that the shortest of all possible ways is the 
distance, i.e. the straight line between the given points, is a 
judgment which, as regards its matter, is perfectly identical, 
and merely regards the same object from different aspects. 
Nor again can the arithmetical judgment, 7 + 5 = 12, be 
synthetical because 12 is not contained in either 7 or 5 : 
the complete subject does not consist in either of these 
quantities singly, but in the combination of them required 
by the sign of addition ; but in this combination, if the 
equation is correct, the predicate must be wholly contained ; 
the equation would be false if some unknown quantity had 
to be added to 7 + 5 in order to produce 12. Here too, 
then, we have a perfectly identical judgment as regards its 
matter, and it is only synthetical formally because it exhibits 
the same number 1 2 first as the sum of two other quantities, 
and then as determined by its order in the simple series of 
numbers, I must now add that it is impossible to express 
everything satisfactorily all at once : what it really means, 
and how it is possible, that thought should represent the 
same matter under different forms, we shall very soon have 
occasion to consider ; and subsequently it will appear that 
my late remarks were not intended to charge Kant with a 
logical oversight so easily detected. 

59. So far our result seems to be this : categorical judg- 
ments of the form '*$* is P' are admissible in practice 
because they are always conceived in the sense which 
we have called particular, and as such are ultimately 
identical. No % one however will feel satisfied with this 
conclusion : it will be rightly objected that it does away 
with the essential character of a judgment, which is that 
it expresses a coherence between the contents of two ideas. 


In fact, if, by the supplementary additions which we spoke 
of, we make our examples into identical judgments, and 
thus compress their whole content into their subjects, so 
that A means the black man, B the drinking dog, C Caesar 
crossing the Rubicon, all that they say, except the barren 
truth that A A, B B, C C, is reduced to this, that 
A exists as a fact continually, B sometimes, and C has 
occurred once in history. In other words, these judgments 
no longer assert any mutual relation between the parts 
of their content^ but only that this content as a composite 
whole is a more or less widely extended fact, and this is 
clearly a relapse to the imperfect stage of the impersonal 
judgment. The following consideration will make us still 
more sensible of this defect. I just now described B as the 
concept of the drinking dog, but properly I had no right* to 
do so ; for this expression, which joins ' drinking ' in the 
form of a participle to the subject ' dog,' is itself only 
conceivable and admissible on the assumption that the 
mark of drinking, P, which is not contained in the subject 
S, can really be ascribed to that subject in a categorical 
judgment, and ascribed to it in the sense of its property or 
state. Now just this possibility has been done away with 
by our previous explanation ; all that it is now competent 
to us to do is to understand B as the coexistent sum of its 
marks a b c d, and to say, this a b c d^ which according to 
the principle of identity is always self-same, has a certain 
reality, while another aggregate of marks, a b c e, has a 
similar reality on another occasion. But we have no right 
whatever to regard the common group a, b, c, as something 
inwardly connected, and more connected in itself than with 
the varying elements d and e, still less as something which 
offers a support to these changing elements as subject to 
attributes. In language, indeed, we should continue to 
describe this a b c as ' dog,' a b c d as ' eating/ and a b c e as 
* drinking dog ' ; but these expressions would rest upon no 
logical ground ; none of our judgments could express any- 


thing but simple or composite perceptions, and between 
the several perceptions, or even the several parts of each 
composite perception, there could be no expressible con- 
nexion such as could show their mere coexistence to be due 
to inner coherence. 

60. Against such a complete failure in its logical purpose 
the mind guards itself, by further transforming the particular 
judgment in a way which may be primarily considered as 
a simple denial that the material of our ideas is thus dis- 
integrated into merely isolated coexistent facts. The addi- 
tions by which we supplemented the subject expressed in 
the categorical judgment, were the means by which we 
helped that judgment to justify itself before the principle of 
identity ; they are now recognised as being also the valid 
ground of fact which qualifies S for assuming a predicate P, 
which, so long as it stood alone, would not belong to it. 
The accessory circumstances, through which S first became 
the true subject 2 of a then identical judgment, appear now 
as the conditions , by the operation or presence of which S is 
so influenced that a P, which before was strange to it, now 
fits and belongs to it consistently with the principle of 
identity. It is therefore the hypothetical judgment which 
takes its place as the second member in this second group 
of the forms of judgment ; it is compounded of a protasis 
and an apodosis, which in the simplest typical case have 
the same subject S, but different predicates, in the protasis 
a Q which expresses the condition accruing to S, in the 
apodosis a P which expresses the mark produced in 
by that condition. All hypothetical judgments with different 
subjects in their two members are abbreviated expressions, 
and can be reduced by easily supplied links to this original 
form, ' If S is Q, S is P. 9 If it is further wished to imply 
that the protasis, which as such is only problematical, is 
actually true, we get the form, 'Because S is Q, S is P 9 : 
and lastly, the assertion that Q is not the ground for *S*s 
being P gives rise to the last form which we need mention, 


Although S is (>, yet S is not PS Logically there is no- 
thing peculiar in these two forms. 

61. This short survey is quite sufficient to characterise 
the external forms of the hypothetical judgment. But an 
observant reader must ask at this point, what right had we 
to translate those supplementary additions, to which the 
true subject 2 of the then identical judgment owed its 
origin, into conditions , which, by operating upon an already 
existing subject *S, give a ground for the predication of P. 
The principle of identity merely asserts the sameness of 
everything with itself; the only relation in which it places 
two different things is that of mutual exclusion. If then we 
supposed various simple elements a b c p q existing together 
in some real form, but without being in any way inwardly 
connected, some of these elements might equally well occur 
at any subsequent moment in any other combination with 
any other element, and the fact of our observing a b c q 
a second time would not enable us to conclude that/ must 
be there too; any r or s might with equal right take its 
place. On the other hand, if we make the quite general 
presupposition that the totality of things thinkable and real 
is not merely a sum which coexists but a whole which co- 
heres, then the law of identity has wider consequences. 
The same a b c q, with which p has once been found in 
combination, can then according to the law of identity 
never be found in combination with a non^, nor can this 
abcq ever occur without its former predicate/. How such 
a cohesion between different elements is conceivable, we 
will leave for a moment an open question; but if it exists, 
it must exist in an identical form in every recurrent instance, 
and (confining ourselves to a combination of three elements) 
given a b> c is the only new element which can necessarily 
accrue, given a c^ b^ and given be, a\ in other words, which- 
ever of these elements occurs first in any case has in the 
second the sufficient and necessary condition for the possi- 
bility and necessity of the accession of the third. That 


element or group of elements to which we here give the 
first place, appears to us then logically as the subject ; that 
which we place second, as the condition which operates 
upon this subject, while the third represents the conse- 
quence produced in the subject by the condition. I wish 
further expressly to point out that this choice of places is 
quite arbitrary, and in practice is decided by the nature of 
the object and our interest in it : in itself, every element in 
such a combination is a function of the rest, and we can 
pass inferentially from any one to any other. It is usual to 
conceive of a number of elements which frequently recur 
together as a subject S, which generally signifies a thing or 
permanent reality: on the other hand, a single element b, 
which is absent in some observations of and present in 
others, is conceived as the accessory condition Q, and a <r, 
which always accompanies <, as the consequence P of which 
Q is the condition. But it is obvious that we may proceed 
in a different way; and in fact mechanical physics are able 
to treat the single and uniform force of gravity, b or <2, as a 
subject, and to investigate the various consequences, P, 
which accrue to it if the bodies upon which it acts (amn 
= S or amr=S l ) be regarded as different conditions to 
whose influence it is liable. 

62. In this way the interpretation by which we arrived at 
hypothetical judgments may be said to be so far justified, as 
that it has been traced back to the most general assumption 
of a coherence between the various contents of thought. 
To prove further than this the admissibility and truth of 
that assumption itself, cannot be part of our undertaking ; 
any such attempt would obviously imply what had to be 
proved, for how could we show that it is permissible and 
necessary to conceive the matter of experience as a web of 
reasons and consequences, if we did not base this assertion 
itself upon a reason of which it was the consequence ? This 
idea of the coherence of the world of thought must therefore 
either be apprehended with immediate certitude, as the 


soul of all thinking, or we must give it up and along with it 
everything that depends upon it. On the other hand, we 
are justified in desiring further elucidation of the possibility 
and the meaning of such a coherence of different elements. 
The possibility of mutual relations between what is different 
is not really threatened by the principle of identity, accord- 
ing to which each thing is related only to itself; for all that 
this principle can affirm is the content of the thing itself; it 
cannot exclude other contents which do not conflict with it. 
But as regards the meaning of the coherence, we must dis- 
tinguish two questions. In logic as here conceived we do 
not trouble ourselves at all as to what the real process may 
be through which the unknown reality, which we express 
well or ill through our ideas, reacts upon itself and produces 
changes in its conditions ; to reflect upon the bond of this 
connexion is the function of metaphysic, and the question 
should find solution in a theory of the efficient cause. Logic, 
on the other hand, which includes in its consideration the 
relations of the merely thinkable which has no real existence 
in fact, is confined to developing the other principle, that of 
sufficient reason ; it has merely to show how, from the com- 
bination of two contents of thought, S and <2, the necessity 
arises of thinking a third, P, and this in a definite relation 
to *$"; if then it were found in actual experience that such a 
union of S 1 and Q l is an accomplished fact, the particular 
consequence P 1 , which according to the necessity of thought 
must follow such a combination in distinction from P' 2 
which could not so follow, could be inferred according to 
the principle of sufficient reason ; but how it comes about 
that the very P 1 , which is required by thought, occurs in 
reality as well, is a question which would be left to the 
metaphysical enquiries referred to. 

63. The law of sufficient reason^ with which we now con- 
clude as the third member and the net result of this second 
group of the forms of judgment, much as it has been talked 
about, has had the curious fortune never to have been, pro- 


perly speaking, formulated, even by those who most fre- 
quently appealed to it. For the ordinary injunction, that 
for every statement which claims validity we must seek a 
ground for its validity, forgets that we cannot seek for that 
of which we do not know wherein it consists ; clearly the 
first thing that has to be explained is, in what relation 
reason and consequence stand to each other, and in what 
sort of thing consequently we may hope to discover the 
reason of another thing. I shall make my meaning clear 
in the shortest way, if, on the analogy of the expression ot 
the principle of identity, A = A, I at once give the formula 
A + B = C as the expression of the principle of sufficient 
reason, adding the following explanation. Taken by them- 
selves, A only =: A, B = B\ but there is no reason why 
a particular combination A 4- B, the very different sense 
of which in different cases is here represented by the 
sign of addition, should not be equivalent to, or identical 
with, the simple content of the new concept C. If we 
thus call A -h B the reason and C the consequence, 
reason and consequence are completely identical, and the 
one is the other ; in this case we must understand by 
A + B any given subject along with the condition by 
which it is influenced, and by C, not a new predicate which 
is the consequence of this subject, but the subject itself in 
its form as altered by the predicate. In ordinary usage 
this is expressed differently. Inasmuch as, in speaking ol 
real facts, the one part A is usually already given, while the 
other B is a subsequent addition, it is customary to describe 
the condition B^ which forms only a part of the whole 
reason A -f #, as the reason in general which acts upon 
the passive subject A ; by C is then usually understood 
nothing but the new property conditioned by B> and this is 
called the consequence ; at the same time, however, the 
property is never thought of as existing on its own account, 
as if in empty space, but as attaching to the subject A upon 
which B was supposed to act. Ordinary usage, therefore, 


though it employs a different nomenclature, means the 
same as I do. If with the idea of powder, A, we connect 
the idea of the high temperature of the spark, B, and thus 
substitute B for the mark of ordinary temperature in A, 
then A + B really is the idea C of exploding powder, not 
of explosion in general ; the ordinary usage makes the high 
temperature B seem to supervene on the given subject A as 
a reason from which the explosion C follows ; but of course 
it conceives this consequence, not as a process which takes 
place anywhere, but as an expansion of the particular 
powder upon which the spark acted. It is not necessary to 
continue such simple explanations any further. 

64. If we consider the whole of our knowledge, we see 
at once that the principle of identity cannot be its only 
source. Taken alone it would isolate every judgment and 
even every concept, and would not open any way to a pro- 
gress from the barren self-identity of single elements of 
thought to their fruitful combination with others. It is a 
mistake, as is sometimes done, to represent this single 
principle as 4:he basis of the truths of mathematics ; the fact 
is that here too it is only the principle of sufficient reason 
which helps to real discovery. From a self-identical major 
premiss nothing new could flow, unless it were possible in a 
number of minor premisses to give the same quantity C 
innumerable equivalent forms, at one time = A + J3 y at 
another = M -f N^ at another = N R ; or, to express 
the same thing otherwise, unless the nature of numbers 
were such that we can divide them all in innumerable ways 
and compound them again in the most manifold combina- 
tions ; and again, unless the nature of space were such 
that every line can be inserted as a part or otherwise 
coherent member in innumerable figures in the most 
various positions, and that each one of the expressions for 
it, which flow from these various relations, is the ground 
for new and manifold consequences. I need hardly men- 
tion that mechanics and physics also make the most 


extensive use of this analysis and composition of given 
facts, and that the process of thought in discovery in these 
branches of knowledge rests upon operations which all 
ultimately come back to the typical formula, A + B = C. 
To Herbart belongs the credit of having brought within the 
ken of formal logic the importance of a mode of procedure 
so prominent in all scientific practice. 

66. Reserving further illustrations for applied logic, I 
have another remark to make about the justification of the 
principle of sufficient reason itself. We were only able to 
show that an extension of our knowledge is possible if 
there is a principle which allows us to make A + B = C. 
We might accordingly attempt to assert the validity of this 
principle at once, as an immediate certitude, like the prin- 
ciple of identity. This is what we have done ; still there is 
a noticeable difference between the two principles. The 
principle of identity expresses of every A an equality with 
itself which we feel immediately to be necessary, and the 
opposite of which also we feel with equal conviction to be 
impossible in thought. The principle of sufficient reason 
lacks this latter support ; we do not by any means feel it 
impossible to suppose that, while every content of thought 
is self-identical, no combination of two contents is ever 
equivalent to a third. The validity of the latter principle, 
therefore, is of a different kind from that of the former ; if 
we call the one necessary to thought because of the im- 
possibility of te opposite, the other must be considered 
rather as an assumption which serves the purposes of 
thought, an assumption of mutual relatedness in thinkable 
matter the truth of which is guaranteed by the concentrated 
impression of all experience. 

I wish not to be misunderstood in this last phrase. In 
the first place, I do not mean that it is a comparison of 
what we experience which first leads the mind to conjecture 
the validity of such a principle ; the general tendency of 
the logical spirit, to exhibit the coexistent as coherent, 


contains in itself the impulse, which, independently even 
of all actual experience, would lead to the assumption of 
a connexion of reasons and consequences. But that this 
assumption is confirmed, that thought does come upon 
such identities or equivalences between different elements 
in the thinkable matter which it does not make, but 
receives or finds, this is a fortunate fact, a fortunate trait in 
the organisation of the thinkable world, a trait which does 
really exist, but has not the same necessity for existing as 
the principle of identity. It is not impossible to conceive 
a world in which everything should be as incommensurable 
with every other thing as sweet is with triangular, and in 
which therefore there was no possibility of so holding two 
different things together as to give ground for a third : it is 
true that, if such a world existed, the mind would not 
know what to do with it, but it would be obliged to recog- 
nise it as possible according to its own judgment. I will 
add further that, when I speak of a kind of empirical 
confirmation of the principle of sufficient reason, I do 
not mean such a confirmation as the whole of our world 
of thought, already articulated in accordance with that 
principle, might find in the fact that external reality, so 
far as it is observable, corresponds with this articulation; 
I am speaking here only of the fact that the thinkable 
world, the contents of our ideas which, whatever their 
source, we find in our inner experience, do really con- 
form to the requirement that they should cohere as reasons 
and consequences. In this stage of logic it is quite in- 
different whether or not there is anything which can be 
called external world or reality besides the ideas which 
move within our consciousness ; like that reality, this in- 
ternal world itself, with all that it contains, is not made 
by thought; it is a material which thought finds in us to 
work upon, and it is therefore for the logical spirit and 
its tendency an object of inner experience; this, then, is 
the empirical object which, by responding to the logical 


tendency and making its realisation possible, substantiates 
the principle of sufficient reason, not as a necessity of 
thought, but as a fact. 

66. As to the nature of this responsiveness in the 
world of thought (if that question is to be raised again 
here), the shortest way to recall it is to observe that the 
position occupied in the system by the principle of sufficient 
reason, as the second law of thought, is analogous to that 
of the act which 1 we placed second in treating of con- 
ception. The possibility of forming general concepts de- 
pended on the fact, not in itself a necessity of. thought, 
that every idea is not incommensurable with every other, 
but that, on the contrary, colours, tones, and shapes 
group themselves in series of cognisable gradations; that 
further there are oppositions of varying degree, as well 
as affinities, in the world of thought, and that opposites 
cancel one another; and lastly, and most important of 
all, that there is a system of quantitative determinations 
enabling us to compare the members of different series, 
which as such stand in no mutual relation. With this 
brief indication, we leave the principle of sufficient reason 
as the conclusion and net result of the second group of 
the forms of judgment. 

C. The General Judgment. The Disjunctive Judgment. 
The Dictum de omni et nullo and the Principium ex- 
clusi medii. 

67. It remains to determine in each particular case, 
What A, combined in what form with what B^ forms the 
adequate reason of what C. This question of fact logic 
must leave to experience and the special sciences; but 
a new question is developed which logic itself must deal 
with. There would be little result from all the activity 
of our mind if we were really obliged in every particular 
case to renew the question to experience, What A, B, and 

1 [See above, 19.] 


C in this instance cohere as reason and consequence? 
There must be at any rate a principle which allows us, 
when once the one truth A+It=Cis given, to apply it to 
cases of which experience has not yet informed us. What 
we are here looking for is easy to find, and has been already 
mentioned incidentally. Whenever we regard A 4- B as the 
reason of a consequence C, we necessarily conceive the 
connexion of the three as a universal one ; A -f- B would 
not be a condition of C, if, in a second case of its occur- 
rence, some casual D instead of C might possibly be found 
combined^ with it. The significance of this in its present 
application is as follows : everywhere, in every subject 
in which A -f B is contained as a mark along with other 
marks, N O P, this A 4- B gives ground for the same con- 
sequence C\ and this C will either actually occur as a 
mark of S, or, if it does not occur, it can only be hindered 
because the other marks, N+ O or N+P or O + P, formed 
together the ground for a consequence opposed to and 
destructive of C; taken by itself, without this hindrance, 
the power of A -f B to condition C never loses its effect. 
If now we conceive A -f- B under the title M as a universal 
concept to which S is subordinate, we may give the fol- 
lowing preliminary expression to the principle just discovered, 
viz. that by right of pure logic and without appeal to 
experience every subject may have that predicate affirmed 
of it which is required by the generic concept above it. 
And it is clear without further explanation that this very 
idea of the subordination of the individual to the universal 
is the comprehensive logical instrument, of which we avail 
ourselves whenever we want to carry further the work of 
thought upon the material given in experience. 

68. The form of judgment, the first of this third group, 
in which the mind expresses this conviction, is that of the 
quantitatively undetermined proposition, in which the place 
of the subject is filled simply by a universal or generic 
concept M\ 'man is mortal,' 'sin is punishable.' I dis- 

Locic, VOL. I. H 


tinguish these as general judgments from the universal 
ones, 'all men are mortal/ t every sin is punishable/ 
Although the fact contained in both forms is the same, 
the logical setting of it in the two cases is quite different 
The universal judgment is only a collection of many 
singular judgments, the sum of whose subjects does as 
a matter of fact fill up the whole extent of the universal 
concept ; thus the fact that the predicate P holds good of 
all M follows here only from the fact that it holds good 
of every single M] it may however hold good of each 
M for a special reason which has nothing to do with 
the universal nature of M. Thus the universal proposition^ 
'all inhabitants of this town are poor,' leaves it quite 
uncertain whether each inhabitant is made poor by a 
particular cause, or whether the poverty arises from his 
being an inhabitant of this town ; so too the universal 
proposition, 'all men are mortal/ leaves it still an open 
question whether, strictly speaking, they might not all live 
for ever, and whether it is not merely a remarkable con- 
catenation of circumstances, different in every different 
case, which finally results in the fact that no one remains 
alive. The general judgment on the other hand, 'man 
is mortal/ asserts by its form that it lies in the character 
of mankind that mortality is inseparable from every one 
who partakes in it. While therefore the universpl judg- 
ment merely states a universal fact, and is therefore only 
assertorial, the general judgment lets the reason of its 
necessary truth be seen through it, and may thus, in the 
sense laid down above 1 , be called apodeictic. This dis- 
tinction of the two forms of judgment will not lead to 
any unheard-of discoveries ; but in comparison with the 
many unprofitable distinctions which encumber logic it 
deserved an incidental mention. It is scarcely necessary 
to remark that in the general judgment it is not the generic 
concept M, occupying the place of subject in the sentence, 
1 [Above, 42.] 


which is the true logical subject of the judgment; it is 
not the universal man who is mortal, but the individual 
S 1 who participates in this type, which in itself is immortal. 
From this we see that the general judgment is properly 
an abbreviated hypothetical judgment; in its full form it 
ought to stand, ' If is M, S is P,' ' If any S is a man, 
this 5 is mortal.' And this justifies us in not introducing 
it in our system until after the hypothetical. 

69. But it is no less clear that we must make another 
step. So long as a universal generic concept M occurs 
as formaHy the subject in the general judgment, the pre- 
dicate P which is joined to it can only be understood 
with equal universality. If we say, 'man is mortal,' the 
predicate embraces all conceivable kinds of mortality, and 
does not determine either the manner or the moment of 
death; or if we say, ' bodies occupy space,' it remains 
unexpressed with what degree of density and of resistance 
each single body realises the universal property of its class. 
But we saw that it is individual men and individual bodies 
which are the real subjects of the general judgment; it 
is therefore quite false to say that P, the mark of their 
class, is a predicate of the individuals in the same universal 
sense in which it is joined in thought (and that not as 
a predicate) to the concept of the class; the truth is 
that P can only occur in each one of these individuals 
in one of the definite forms or modifications into which 
the universal P can be analysed or particularised. The 
mind corrects this mistake by means of the fresh assertion, 
' If any S is an M^ this S is either / T or / 2 or /Y and 
here/ 1 / 2 / 3 mean the different kinds of a universal mark P 
which is contained in the generic concept M. This is 
the familiar form of the disjunctive judgment, the second 
in this third group, and one which, as such, requires no 
further explanation. It is usual to mention along with it 
the copulative judgment (' S is both / and q and r'\ and 
the remotive judgment ('-S* is neither/ nor ^-nor r')' t but 

H 2 


in spite of the external analogy of form, neither of these 
has the same logical value as the disjunctive; the first 
is only a collection of positive, the second of negative, 
judgments with the same subject and different predicates, 
which latter are not placed in any logically important 
relation to each other. The disjunctive judgment alone 
expresses a special relation between its members : it gives 
its subject no predicate at all, but prescribes to it the 
alternative between a definite number of different pre- 

70. The thought expressed by the form of* the dis- 
junctive judgment usually finds utterance in two separate 
laws of thought, the Dictum de omni et nullo and the 
Principium exclusi tertii inter duo contradictoria ; but the 
amalgamation of them in a single third law is not only 
easy but necessary. The careless formulations often given 
of the first are completely false, e.g. 'What is true of 
the universal is true also of the particular/ 'What is true 
of the whole is true also of the parts'; on the contrary, 
it is self-evident that what holds good of the universal 
as such or of the whole as such, cannot 'hold good of 
the individual as such or of the parts as such. The only 
correct formula is, quidquid de omnibus valet valet etiam de 
quibusdam et de singulis, and quidquid de nullo valet nee de 
quibusdam valet nee de singulis. But this form of expression 
(for the history of which see Rehnisch, Fichte's Zeitschrift, 
Ixxvi, i) is as barren as it is correct ; for to hold good of 
all is and means from the very first nothing else than to 
hold good of each one ; if therefore anything worth saying 
is to take the place of this bare tautology, the nature of the 
universal concept must certainly be substituted for the mere 
sum of alL But in that case the principle cannot really be 
accurately expressed except in a form which means precisely 
the same as the disjunctive judgment ; viz. whenever a 
universal P is a mark in a universal concept M 9 one of its 
modifications, /W 3 , to the exclusion of thfe rest, belongs 


to every S which is a species of M\ whenever a universal 
P is excluded from a concept AT, no one of the modifica- 
tions of P belongs to any S which is a species of M. 

71. Of this complete law of thought the ordinary ex- 
pression of the dictum de omni et nullo only regards >the one 
and positive part, which, as we saw, cannot by itself be 
accurately expressed, the general idea, namely, that the 
particular is determined by its universal : the other and 
negative part, which defines the manner of this determina- 
tion, the idea that the particular admits only one specific 
form of its generic predicate to the exclusion of the others, 
has found only a partial expression in the principle of the 
excluded middle. I think that I can say what I have to 
say about this most simply as follows. Suppose a subject S 
subordinate to M^ and that this subordination implies that 
S must choose its own predicate from amongst/ 1 /'/ 3 , the 
specific forms of P, a universal mark belonging to J/, then, 
if there are more than two of these forms, the affirmation of 
one of them as predicate of S will involve the negation of 
all the rest, but the negation of one of them will not 
involve the affirmation of any particular one of the rest; 
what is not / l has still an open choice between /* / 3 / 4 . To 
predicates of this sort it is usual to ascribe the opposition of 
contrariety. If however there are only two specific forms of 
P, /* and / 2 , and S must have a specific form of P for its 
predicate, then not only does the affirmation of one of them 
as predicate of S involve the negation of the other, but also 
the negation of the one involves the definite affirmation of 
the other ; / l and / 2 are then opposed to one another 
contradictorily. Thus for the line (S) 9 which must have 
some direction (P\ straight (/') and crooked (/ 2 ) are contra- 
dictory predicates, and so for man, whose nature it is to 
have sex, are male and female : for any other subjects, of 
which it was not yet established whether their concepts 
contained the universal P at all, these predicates would be 
only contrary; for such subjects the division of their 


possible predicates will be always threefold, they are either 
male, female, or sexless, either straight, crooked, or form- 
less. Now the principle of the excluded middle asserts 
nothing but what we have just remarked, that of two 
predicates which are contradictory for a subject S, S always 
has one to the exclusion of the other, and if it has not the 
one it necessarily has the other to the exclusion of any 
third. So regarded, this law is only a particular case of the 
more universal law of which the disjunctive judgment is 
the expression, viz. that of all contrary predicates whose 
universal P is contained in the generic concept M of 
a subject S, S has always one to the exclusion of the rest, 
and if it has not any given one, it has only left it the choice 
between the others ; this choice becomes a definite affirma- 
tion when it can only fall on one member, i. e. in the 
extreme case where the number of contrary predicates is 
only two. Such a case, which is all that is covered by the 
principle of the excluded middle, is no doubt of peculiar 
importance in practice, but a system of logic can only 
treat it as a particular instance of the more universal prin- 
ciple, which we have already mentioned several times and 
which we will briefly describe as the disjunctive law of 

72. It is usual to represent this differently. From motives 
which are likewise only intelligible on practical grounds, 
the logical desire has arisen to omit the presupposition 
to which we have adhered throughout (viz. that the given 
subject S be already understood to stand in a necessary 
relation to the universal predicate />), and to be allowed to 
speak of two predicates which hold good as contradictories 
of any subject whatever. It is soon found that this is only 
possible, if the aggregate of all conceivable predicates be 
divided into a definite Q and the sum of all those which 
are not Q or non-<2 ; it is then certain that any subject, 
whatever it may mean, must be either Q or non-@, either 
straight or not-straight, for not-straight will include not only 


crooked, but annoying, sweet, future, everything in short 
which lies outside of straight. On this point I may repeat 
what I said 1 about the limitative judgment, viz. that non-(? 
is not a real idea at all, such as can be treated as predicate 
of a subject ; it is only a formula expressing a mentally 
impracticable task, the collection of all thinkable matter that 
lies outside a given concept into a single other concept. 
Moreover there is no real reason for propounding this 
insoluble problem ; everything which it is wished to secure 
by the affirmative predicate non-Q is secured by the 
intelligibly negation of Q. I therefore consider it quite 
improper to speak of contradictory concepts , i.e. concepts 
which are of themselves contradictorily opposed and 
therefore retain that opposition when treated as predicates 
of one and the same subject, whatever that subject may be: 
if we want a contradictory relation which shall hold good 
universally, always, and in regard to every subject, it can 
only exist between two judgments, ' is (?,' ' S is not Q. 1 
Accordingly the precise expression of the principle of the 
excluded middle would be, that of every precisely deter- 
mined subject 6" either the affirmation or the negation of an 
equally determinate predicate Q holds good, and no third 
alternative is possible ; wherever it appears to be possible, 
S or <2 or both have either been taken in more than one 
sense or in an indefinite sense in the first instance, or their 
meaning has been unconsciously or involuntarily changed 
in the course of reflection. 

73. I have one more observation to add. No one doubts 
that the same subject can be at the same time red, sweet, 
and heavy, but that it is red only when it is neither green 
nor blue nor of any other colour, and that it cannot be 
straight and crooked at the same time. Yet it does not 
seem to me to be immediately evident that, as is sometimes 
asserted, the case in which two predicates / l and / 2 are 
incompatible in the same subject is just that in which they 
1 [See above, 40.] 


are contrary species of the same universal P and therefore 
admit of comparison, whereas other predicates / q r are 
compatible in the same subject when, as species of quite 
different universals P Q R, they admit of no comparison. 
On this point I venture the following reflections. Every 
predicate/ 1 of a subject S must be regarded, in accordance 
with what we said above and the formula A + B~C> as a 
consequence of a group of marks A 1 -\-J3 l in S, which group 
tends in all cases (and therefore in the case of S] to produce 
the same result C l (in this case /*). If now the same S is 
to have at the same time the predicate/ 2 , comparable with 
/*, it is easy to understand that/ 2 must depend on a group 
of marks A 2 + .Z? 2 , similarly comparable with A 1 + l , existing 
side by side with the latter in S, and in all cases of its 
occurrence (and therefore in the case of S] giving ground 
for the result C 2 (in this case /'). But the consequence 
of the very comparability of A 1 + B^ and A^ + B* must be 
that, according to a new principle of the general form 
A+B-C, viz. [A l + l ] + [A* + jB']=C\ their meeting in 
the same subject S will' furnish the sufficient reason for a 
new consequence C z , in which the two specific predicates 
p l and p 2 coalesce, and which, as it must resemble both 
of them, we will call /*. The only reason, therefore, why 
two contrary and comparable predicates p 1 and / 2 would be 
irreconcileable, is that they would always give rise to a third 
and simple/ 3 ; on the other hand, two disparate and incom- 
parable predicates / and r, such as sweet and warm, could 
coexist permanently in *$" because there is no principle 
such as (A+) + (M+N)=C enabling the two disparate 
grounds A+B and M+N, on which the predicates re- 
spectively depend, to produce like / l and / 2 a third and 
simple predicate. I will not quarrel with those who find 
the whole of this exposition superfluous ; it seems to me to 
have some point, when I turn from the examples which 
logic traditionally employs to others which it would do well 
not to forget. When anyone says of gold that it is yellow, 


he has, it is true, no occasion to think of this simple 
property as a product of two other imperceptible ones, 
which properly speaking must have been produced separ- 
ately by two conditions coexisting in gold, but could not 
remain separate. But when two motive forces contrary or 
even contradictory in direction act upon a material point, 
that which in the previous case would have been a needless 
assumption is now an actual fact] we have to conceive both 
of the condition which tends to produce the motion p l and 
of that which tends to produce p* as operating at the point, 
and of the two motions themselves as at every moment 
predicates of that point, but predicates which cannot main- 
tain themselves separately but coalesce in a third / 3 , the 
motion in the diagonal. 

And ultimately this is seen to be true in all cases. A 
crooked line may appear indifferently red or green: but 
if the conditions of both appearances were operating at the 
same time and with the same force, it would help us but 
little to assert, on the principle of exclusion, that the image 
of the line cannot have these two contrary properties; it 
must present some appearance. As however these two con- 
ditions are comparable and capable of forming a resultant, 
a third colour will appear, the production of which will 
satisfy the claims of the two conditions, but will at the 
same time contain the reason why the two contrary 
colours, which singly they would have produced, cannot 
exist separately side by side. 

74. The series of judgments concludes here by an inherent 
necessity. The more definitely the disjunctive judgment pre- 
scribes to its subject the choice between different predicates, 
the less can this uncertainty be final ; the choice must be made. 
But the decision, what p^ or / 2 belongs to *$*, cannot come 
from the fact (which is thus far the only fact) that S is sub- 
ordinate to M, for it is just because it is a species of M 
that it is still free to choose : that decision can only come 
from the specific difference by which S, as this species 


of M, is distinguished from other species of it, The pro- 
position < J/(and every S which is M) is PJ must therefore 
have added to it a second proposition which brings to light 
the specific character of S, the particular subject always 
in question, and shows us what species of M it is ; and 
from the union of the two propositions must arise a third, 
informing us what particular modification / of the universal 
P belongs to this S because it is, not only a species of M> 
but this species. The form of thought which combines two 
judgments so as to produce a third is, speaking generally, 
inference^ and it is therefore to the exposition of, inference 
that we have now to pass, 

Appendix on immediate inferences. 

In conformity with tradition I insert some explanations 
here which would more correctly come under the head 
of applied logic. Of the same subject and the same 
predicate P the universal affirmative judgment, A, asserts 
* All S are P] the particular affirmative, /, ' Some are PJ 
the universal negative, E, ' No S is P,' and the particular 
negative, O, ' Some are not P.* The question is, what 
immediate inferences can be drawn from the truth or un- 
truth of one of these four judgments in regard to the truth 
or untruth of the other three ? From the Dictum de omni 
et nullo and the principle of the excluded middle t we ob- 
tain the following results. 

75. Between each universal judgment and the particular 
of like name there is the relation of subalternation. Going 
from the universal to the particular or ad subalternatam^ we 
infer the truth of the latter from that of the former, but 
from the untruth of the universal we cannot infer either the 
truth or the untruth of the particular. The correctness 
of the first inference is obvious at once, and it only requires 
the removal of a misunderstanding to make the impossibility 
of the second equally so. A person who denies the uni- 
versal proposition, ' all are P, } is usually led to do so by 

chap, no * AD SUBALTERN ANTEM: 107 

having already observed some 6" which are not P ; but he 
will not have included all S in this observation. His in- 
tention therefore generally is merely to deny the universal 
application of the proposition to all S, while leaving its 
truth in single cases of S undisputed ; and thus it is that 
in ordinary speech expressions such as ' It is not true that 
all S are also /Y are actually understood to admit inci- 
dentally the truth of the particular proposition, * some S are 
P.* Logic, on the other hand, knows nothing of these 
unexpressed suggestions in the denial of the universal 
proposition : it recognises merely what lies in the expressed 
negation itself. But it is just this which is ambiguous. 
For the asserted untruth of the proposition, 'all are PJ 
is equally a fact, whether the proposition is true of only some 
S or of none. So long therefore as this ambiguity is not 
removed by accessory statements, we cannot infer from the 
negation of the universal proposition either the truth or the 
untruth of the particular. 

76. Going in the opposite direction, from the particular 
to the universal or ad subalternantem, we infer the un- 
truth of the universal judgment from that of the particular, 
but not the truth. Here, too, the first conclusion is obvious, 
if we avoid the ambiguity already alluded to. A person 
who denies the proposition, 'some S are P, 1 may, it is 
true, ; ntend merely to deny that P is confined to some 
S, and the effect of his meaning that ' not only some S are 
P ' would then be to affirm the universal proposition ' all 
S are P. 9 But just because this consequence would directly 
imply that the particular judgment, ' some S are P 9 * also 
remained true, logic cannot possibly interpret the denial 
of that judgment in this way. For logic this denial means 
nothing but that f there is no such thing as some -S" which 
are P 9 ; and what is not even true in some cases is still less 
true in all. Consequently the negation of the particular 
always negates the universal too. The impossibility of the 
second inference explains itself; the truth of P in the case 


of some can never prove its truth in all S: it is only 
because this unjustifiable generalisation of single obser- 
vations is the commonest of logical mistakes, to which 
science and culture owe most of their errors, that it is 
worth while to prohibit with especial emphasis this false 
inference ad subalternantem. 

77. Universal judgments are contradictorily opposed to 
particulars of unlike name, A to O and to I and vice 
versa ; we infer ad contradictoriam both the untruth of 
the one from the truth of the other and the truth of the 
one from the untruth of the other. The first inference 
needs no explanation, the second a brief one. If we 
deny the proposition A, 'all are P,' the denial is con- 
sistent with both the assumptions E and O, ' no S is P,' 
and ' some are not P' ; but the second, which is included 
in the first, is true in any case; consequently the truth 
of O follows certainly from the untruth of A. If we 
further deny (9, ' some S are not PJ this means, according 
to what we said above, * there is no such thing as some 
S which are not P,' and this is equivalent to A, 'all 
S are P. 9 If we deny E^ ' no is P,' either all or some S, 
in any case the latter, are P, and consequently / is true, 
'some S are P' : if we deny /, this means, 'there is no 
such thing as some S which are P,' and is equivalent to 
the affirmation of J, ' no is P.' 

78. The two universal judgments of unlike names are 
only contrariwise opposed, and we infer the untruth of 
the one from the truth of the other, but not the truth 
of the one from the untruth of the other. The first case 
is obvious: the impossibility of the second follows, after 
what we said before, from the consideration that, while 
the negation of a universal judgment allows an inference 
ad contradictoriam to the truth of the particular of unlike 
name, the truth of the latter does not allow an inference 
ad subalternantem to that of the universal to which it is 
subordinate. Lastly, the relation between the two par- 


ticular judgments / and O is called subcontrary opposition. 
We infer ad subcontrariam the truth of the one from the 
untruth of the other, but not the untruth of the one 
from the truth of the other. In fact, the two propositions, 
4 some S are not P] and c some *$* are /Y ma Y both be 
true together; but if one is denied, the truth of the op- 
posite universal follows ad contradictoriam, and from this 
again follows ad subatternatam the affirmation of the par- 
ticular subordinate to it. 

79. I may also mention another logical operation which 
has a kindred object. All observations, which always 
admit ultimately of being expressed in the form of a 
judgment ' $ is /y present us only with that combination 
of S and P which actually occurs at the moment of 
observation : they tell us nothing as to whether S and P 
will be separable or not in other cases, whether, in fact, 
there are S which are not P or P which are not S. 
Now we have a practical interest in this question which 
is very intelligible : we want to know whether a P which 
has occurred in S may be considered as a mark> enabling 
us to determine the nature of the subject in which it 
occurs : in short, whether everything which has the cha- 
racteristics of a P is also always an S. The answers to 
be expected to this question will accordingly take the 
form, ( P is *$"'; and they are therefore called conversions 
of the original judgments which gave rise to them. It is 
also obvious that we have a special interest in knowing 
whether P points to a subject *$* necessarily and always, 
or only possibly and sometimes ; whether, as it is ordinarily 
put, all P y or only some, are $". Hence it is the quantity 
of the original and the converted judgment to which par- 
ticular attention is paid, and the conversion is called pure 
(conversio pura) when the quantity of the second is that of 
the first without any change, and impure (conversio impura) 
when it is different, especially when the universal truth 
of the original judgment has to be reduced to particular, 


in order to make it true when converted. The results 
are as follows. 

80. The universal affirmative judgment, 'all S are P 9 
understands by P either a higher genus in which is 
contained along with other species, or a universal mark 
in which S partakes along with other subjects. In both 
cases there is a part of P left which has nothing to do 
with S, and only impure conversion can take place into 
the particular judgment ' some P are 6V This rule deserves 
attention, for it is one of the commonest mistakes of 
carelessness and one of the most favorite means of de- 
ception to substitute the universal for the particular infe- 
rence, and to assert, ' If P belongs to all S, then S belongs 
to all P.' It is true that we do meet with universal 
affirmative judgments which admit of this pure conversion, 
those viz. in which the extents of S and P exactly cover 
each other, and P therefore belongs not only to all S, 
but only to all S, so that all P are also S. Such so-called 
reciprocal judgments are, 'all men are naturally capable 
of language,' 'all equilateral triangles are equiangular'; 
they can be converted into, 'all that is naturally capable 
of language is man,' 'every equiangular triangle is an 
equilateral one.' But it is only knowledge of the matter 
of fact contained in the judgment in question which can 
assure us that the relation, upon which this possibility 
depends, holds good between S and P in any particular 
instance. Mathematics, therefore, where the pure con- 
version of universal affirmative judgments is frequent, are 
right in demanding special proof in every case of the 
truth of the converted judgment, and by this caution 
inculcate the rule that by right of mere logic the universal 
affirmative judgment admits only impure conversion into 
a particular affirmative. It is otherwise with the universal 
negative judgment, 'no S is P. 9 This complete exclusion 
of the two concepts from each other clearly holds good 
reciprocally, and justifies the assertion that 'no P is S. 9 


The universal negative judgment is therefore convertible 
into another universal negative. 

81. The particular affirmative proposition, ' some are 
/Y obviously yields pure conversion into another particular 
affirmative, ' some P are S? And this inference is satis- 
factory in all cases in which P is a universal predicate in 
which S partakes along with other subjects ; thus the asser- 
tion, 'some dogs bite,' is rightly converted into 'some things 
that bite are dogs.' But when S is the genus of which P is 
a species, as in the proposition, ' some dogs are pugs/ the 
only logically admissible conversion, ' some pugs are dogs/ 
will contrast unfavourably with the actually true one, ' all pugs 
are dogs.' The former is no doubt true also, but it ex- 
presses only a part of the truth, and in a form which 
appears rather to deny than to affirm the other part, that all 
other pugs also are dogs. We feel this still more if we start 
with the judgment, { all pugs are dogs,' and convert it twice 
over. From the first conversion, ' some dogs are pugs,' we 
cannot get back again by the second to the original proposi- 
tion ; and thus the logical operations have here resulted in 
eliminating a part of the truth. This inconvenience could 
easily be avoided if the expressions of quantity were re- 
garded, as the sense requires that they should be, as 
inseparable froni their substantives ; we should then formu- 
late the proposition, in the first instance as follows, ' all pugs 
are some dogs'; then byconversion, 'some dogs are all 
pugs,' and by a second conversion, ' all pugs are some dogs.' 
But it is not worth the trouble to improve what are after 
all barren formulae. 

The particular negative judgment, ' some S are not P,' as 
such asserts merely the separability of S from P, not that of 
P from S also. The pure conversion therefore, ' some P 
are not -5V does not hold good universally, but only of those 
P which are predicates common to different subjects, and 
are not therefore exclusively dependent upon the nature of 
S for their occurrence. For this reason the proposition,, 


4 some men are not black,' can be converted into, < some- 
thing black is not man'; but the judgments, 'some men are 
not pious, 5 * some are not Christians/ would yield ' some- 
thing pious is not man/ ' some Christians are not men,' both 
inadmissible because piety and Christianity, though not 
belonging to all men, belong only to men. These dis- 
advantages are in general only avoided by joining the nega- 
tion to the predicate, and then converting the proposition, 
' some S are non-/Y like a particular affirmative into ' some 
non-P are S ' ; e. g. * something not-black, something not- 
pious, some non-Christians, are men.' 

82. The process necessary in this case has been extended 
to all judgments under the name of conversion by contra- 
position : in the affirmative judgments the negation of non- 
P takes the place of the affirmation of P, in the negative the 
affirmation of non-P takes that of the negation of P; the 
judgments thus changed are then converted according to 
the ordinary rules. In this way we get the following results ; 
first, for A, ' all S are P,' ' no 8 is non-P,' and so ' no non- 
P is S ' ; for /, on the other hand, * some S are P,' the 
transformation into, 'some S are not non-P,' would not, 
after what has been said above, allow any conversion, and 
contraposition would therefore be impossible ; for , again, 
' no is P/ we get ' all S are non-P,' ' some non-P are .' 
To carry out these operations in actual instances would pro- 
duce unshapely and unnatural forms of expression ; the 
substantial meaning of the four forms of judgment may be 
given more simply by replacing their quantitative determina- 
tions by the equivalent modal ones : even the contraposi- 
tion of /, which in itself is impossible, is thus made avail- 
able. The conversion of A would then mean, 'If the 
predicate P belongs to all individuals of a genus 6", it is 
impossible for anything in which this mark is absent to be 
an S ' : that of / would mean, ' If P is only known to 
belong to some species of S, it is not necessary, but only 
possible, that something in which P is absent should not be 

Chap, no CONVERSION. \ 1 3 

an *': that of E, ' If the mark P is universally absent from, 
or contradictory of, the genus *9, it is not necessary, but 
only possible, that something which similarly lacks or is 
contradicted by P should be a species of S ; ; and the same 
inference applies to O also, ' If some 6" are not P, some- 
thing which also is not P may be an S but need not 
be so/ * 



The Theory of Inference and the Systematic Forms. 

Preliminary remarks upon the A ristotelian doctrine of 

I HAVE pointed out the unsolved problem which compels 
us to advance beyond the disjunctive judgment. Before I 
follow up this thread of connexion systematically, I think it 
will be advantageous to state the theory of syllogism in the 
form which it received from Aristotle. I shall not however 
follow the original exposition of the great Greek philosopher, 
but the more convenient form which came into vogue later. 
The writings of Aristotle are preserved, and anyone who 
takes an interest in the origin of these doctrines may easily 
enjoy his masterly development of them : but when we are 
concerned, not with the history of the thing, but with the thing 
itself, it would be useless affectation to prefer the inconvenient 
phraseology of the inventor to those improvements in detail 
which subsequent ages have placed at our disposal. 

88. Following Aristotle, we give the name of inference 
or syllogism to any combination of two judgments for the 
production of a third and valid judgment which is not 
merely the sum of the two first. Such production would 
be impossible if the contents of the antecedent judgments, 
the two premisses, propositiones praemissae^ were entirely 
different ; it is only possible if they both contain a common 
element M> the middle concept or terminus medius, which 


the one relates to 6", the other to P. This medium brings 
the two concepts -S and P into connexion, and they can 
then meet in the conclusion in a judgment of the form 
* S is P,' or, more shortly, S P, from which the middle 
concept which served to produce it has again disappeared. 
There is no reason in the nature of the case for making a 
difference of value between the two premisses SM and 
P M but 3, tradition, which cannot be disregarded without 
subjecting all established rules to a bewildering change of 
meaning, has decided that the premiss which contains along 
with M the predicate P of the coming conclusion shall 
be called the major premiss, and that which contains S, the 
subject, the minor \ the conclusion itself is always conceived . 
in the form S P, not in the reverse form PS. This 
being presupposed, the further differences in the position 
which the three concepts may assume give rise to the 
following four arrangements, of which the first three re- 
present the three figures of Aristotle, while the fourth 
forms that of Galen. 

(i)MP (2)PM 



84. If we now ask whether, and under what conditions, 
these arrangements of premisses, which are in the first 
instance merely based upon rules of combination, give 
ground for a valid inference, we find at once that S and P 
can only be united in the conclusion if the middle concept 
remains precisely the same; their union is obviously un- 
justifiable as soon as the M connected with *$* in the one 
premiss is different from the M connected with P in the 
other. Such a division of M would give four concepts in 
the premisses, instead of the necessary and sufficient three ; 
the avoidance of this quaternio terminorum^ and the securing 
of complete identity in the middle term, is therefore the 
condition of conclusiveness in all figures alike. To fulfil 

I 2 


this condition it is first of all necessary in all figures to 
exclude any ambiguity in the meaning of the word which 
denotes the middle concept; but besides this there are 
special precautions for the same purpose, which the peculiar 
structure of the several figures renders necessary, and which 
we have now to mention. 

85. In the first figure S is included in M in the minor 
premiss, M in P in the major, and therefore S in P in 
the conclusion. The idea upon which this inference is 
based is evidently that of subsumption; that which is a 
predicate of the genus is a predicate of every subject of 
the genus. This is of itself sufficient to show that the 
major premiss in the first figure must be universal ; for it 
has to express the rule which is to be applied to the 
subject of the minor. The necessity that the middle term 
should be identical leads to the same result. For the S 
of the minor premiss is always a definite kind or a definite 
case of M\ this however is not expressed in the form of 
the proposition ; as far as the form goes S might be merely 
any kind of M in general ; if this indeterminate M is to 
be the same as that which the major premiss asserts to 
be P, this can only be secured if the major premiss speaks 
universally of all M, thus including the indeterminate cases 
along with the rest. It is true that in that case the M 
expressed in the major premiss is not identical with the M 
of the minor, which, as predicate of , necessarily signifies 
only a part of the whole extent of M; but this apparent 
difficulty disappears when we consider that the M of the 
major premiss which is actually employed in producing the 
conclusion is likewise only a part of that which is expressed, 
that part, namely, which is intended in the minor. Further, 
as the inference in the conclusion depends upon the sub- 
ordination of S to -#/", this subordination must be a fact, in 
other words, the minor premiss which expresses it must be 
affirmative; if it were negative, it would simply deny the 
existence of any ground for the validity of the conclusion. 


On the other hand it does not affect the logical connexion 
of the syllogism, but depends merely upon its particular 
content, whether the major premiss affirms or denies P of 
M y and whether the application furnished by the minor of 
the general rule to an instance embraces all S or only some. 
The quality of the major premiss and the quantity of the 
minor are therefore unlimited. Lastly, the relation, whether 
affirmative or negative, in which the major premiss places 
M to P, must be transferred unaltered to the unaltered 
subject, whether universal or particular, of the minor ; the 
conclusion therefore has the quality of the major and the 
quantity of the minor. If we suppose all the possibilities 
exhausted for which these rules leave room, we get four 
valid kinds or moods of the first figure. Their scholastic 
names Barbara^ Celarent, Darii, and Ferio, which by the 
three vowels in order denote (as every one knows) the 
quantity and quality of the premisses and the conclusion, 
show at a glance the distinctive feature of the first figure, 
namely, its capacity to produce conclusions of every kind. 

86. The premisses of the second figure show us two sub- 
jects S and P in relation to the predicate M. If both 
subjects either have or have not this predicate, i.e. if both 
premisses are affirmative or both negative, no inference can 
be drawn from them as to a mutual relation between S and 
P. For innumerable subjects may all participate in, or all 
be excluded from, a mark M, without necessarily having any 
other point in common, and in particular without the one, 
S, being necessarily a species of the other, P. Only if the 
one subject has or has not the mark M always or universally, 
while the other is related to M in the opposite way, is there 
ground for concluding that the second cannot be a species 
of the first. The premisses in the second figure must there- 
fore be of opposite qualities, and one of them must be 
universal. As however it is the tradition that this second 
subject should be supplied by the minor premiss, the pre- 
miss in which the first is mentioned, i.e. the major, must be 


the universal one. Thus the conditions of the second figure 
may be summed up as follows : the major premiss is uni- 
versal, but is not limited as to quality; the minor is of the 
opposite quality to the major and is not limited as to quan- 
tity; the conclusion is always negative, and has the quantity 
of the minor. The possible moods are Camestres, Baroco^ 
Cesare, Festino. 

87. The third figure brings the same subject M into 
relation to two predicates, P and S. If M has both predi- 
cates, i. e. if both premisses are affirmative, the union of P 
and S must be possible, and the conclusion therefore, ac- 
cording to the usual logical expression of such a possibility, 
is, ' some S are P. 9 The necessary identity of M is in this 
case sufficiently secured by the universality of one premiss, 
it does not matter which ; for it clearly makes no difference 
whether all M have the mark P and only some have 6*, or 
whether all Mhzve S and only some P\ in either case there 
are always some M which have both and thereby justify the 
conclusion, which is always particular, * some S are P.* 
Moreover this case, in which M is subject in both pre- 
misses, is just one in which its identity might be easily 
guaranteed by a word of completely individual meaning, 
the proper name of a person for instance. We often meet 
with such inferences : in order to prove the compatibility of 
two actions which seem to be mutually exclusive, we bring 
forward an instance, e.g. * Socrates was P, and Socrates was 
also 5,' consequently ' what is may also be P 9 * or * some 
S 1 is P.' Logic justifies such inferences by attributing to the 
singular judgment, i.e. one whose subject is not an inde- 
finite part of a universal concept but a perfectly definite 
and unique individual, the syllogistic value "of a universal 
judgment. Thus this case comes under the above rule, 
Which, where both premisses are affirmative, requires one to 
be universal, prescribes a particular affirmative conclusion, 
and admits the moods Darapti, Datisi^ and Disamis. 

88. Again, if the same subject possesses one of the marks 


but not the other, i. e. if one premiss is affirmative, the other 
negative, S and P must be separable, or, according to the 
ordinary phraseology, the particular negative conclusion 
follows, * some *9 are not P.' In this case also it is suffi- 
cient for the identity of M that one premiss, it does not 
matter which, should be universal, but the minor premiss 
must be affirmative. For though one of two marks which 
occurs in a given subject is no doubt always separable from 
the other which does not occur in that subject, the latter is 
not necessarily separable from the former ; it is further 
conceivable that if it exist at all it can only do so in 
conjunction with the other. Thus life without intelligence 
is a possible mark of an animal, but not intelligence 
without life. It is therefore the affirmed mark only which 
is separable ; only of it as subject can the conclusion 
assert that it is not always combined with the other as 
predicate; and as this subject of the conclusion is cus- 
tomarily furnished by the minor premiss, the minor premiss 
must be affirmative and only the major can be negative. 
Under this condition mixed premisses yield the moods 
Felapton^ Ferison, and Bocardo, these like the preceding 
ones having only particular conclusions. 

89. Lastly, it is asserted by logic as a universal principle 
that in the third as in the other figures two negative pre- 
misses yield no valid inference. This is incorrect ; a con- 
clusion may be drawn from them similar in kind and equal 
in value to those which are derived from affirmative or 
mixed premisses. For if the first of these prove that and 
P may exist together, and the second that they may exist 
apart, two negative premisses prove with equal ground that 
S and P are not contradictorily opposed, and that accord- 
ingly what is not S need not therefore be P ; in ordinary 
phraseology, 'some not-*S" are not P? I cannot see why 
this conclusion should stand lower in value than the two 
others ; the first only says to us, * when you find S, be pre- 
pared for the possibility of finding P,' the second, ' when 


you meet with 6* do not reckon upon the existence of /Y 
and similarly the third, * where you do not observe S, 
beware of inferring for that very reason the presence of P. 9 
In life we often meet with such inferences ; over and over 
again, when the necessary presence of some quality has 
been over-hastily concluded from the absence of some 
other, we appeal to instances in which neither the one nor 
the other is found, and so correct an erroneous prejudice by 
an inference in the third figure from two negative premisses. 
This conclusion therefore is undoubtedly valid, but it would 
be an anachronism to invent supplementary names for its 
various moods. 

90. The premisses of the fourth figure, ascribed to 
Claudius Galenus, are in form the counterpart of the first 
figure of Aristotle, but do not equal it in value. Its moods 
are Bamalip, Cakmes^ Dimatis, Fesapo, Fresiso. As to the 
premisses of Bamalip, e. g. ' All roses are plants/ * All 
plants need air,' every one who thinks naturally will tacitly 
transpose them, and draw the conclusion of Barbara in the 
first figure, * All roses need air.' It is true that this conclu- 
sion is then of the form PS, but the form SP t which 
is required by the fourth figure, can be easily obtained 
from it by conversion, 'some things that need air are roses.' 
On the other hand .we cannot by conversion recover from 
this conclusion in the fourth figure the one which we drew 
from the same premisses in the first ; its conversion only 
yields the particular proposition, 'some things which are 
roses need air/ Thus in this case the conclusion in the 
figure of Galen actually loses a part of the truth which 
is established by the premisses, a bad recommendation 
for a process of inference, the function of which is always to 
conclude from what is given as much new truth as possible. 
This awkwardness could indeed be avoided, as was shown 
before, but the inference would not thereby be made more 
natural. Equally unnatural are Calemes and Dimatis^ the 
premisses of which will always be transposed by the un- 

Chap. III.] REDUCTION. 121 

sophisticated mind and applied in Celarent and Darii of 
the first figure : they do not indeed occasion a loss of 
truth, since the negative conclusion of Calemes admits 
pure conversion, while that of Darii is particular like 
that of Dimatis. It is only Fesapo and Fresiso which 
are less readily reducible to the first figure, owing to the 
negative minor premiss which results in both and the 
particular major which results in the latter; by pure con- 
version of their majors they can be transposed into Felapton 
and Ferison of the third figure instead, and this change will 
have the same effect of making the conclusions more 
natural. In all points, therefore, the fourth figure is a very 
superfluous addition to the three figures of Aristotle. 

91. Aristotle considered the inferences in all the three 
figures to be valid, but only that in the first to be perfect, 
because in this figure only does the ground upon which 
all inference depends for its possibility, the subordination of 
the particular to the universal, find formal expression in the 
structure of the premisses. In the other figures too, indeed, 
(as he held), the inference rests upon the same principle, 
and the relations of subordination, which are necessary and 
sufficient for drawing a conclusion according to that prin- 
ciple, are contained in the premisses and do not need 
supplementing by information from without ; but they are 
not exhibited in the actual structure of the premisses ; we 
have to look for them there. To make good this formal 
defect in the two latter figures, Aristotle has shown us how, 
without any change of content, their premisses may be 
transformed into those of the first figure. To some people 
this has seemed superfluous, and they have objected that 
the two other figures also conclude according to principles 
of their own and requiring no other evidence : thus the 
fundamental idea of the second, that if two things stand 
in contrary relations to the same mark the one cannot be 
a species of the other, is clear in itself and independent 
of the principle of subordination. I doubt this, but shall 


not pursue the point further; for to hold that the conclu- 
sions of the two latter figures are drawn upon any principle 
at all, is to admit that the ground of all inferences is the 
subordination of the particular to the universal ; for to what 
did those figures apply their principles if not to justify the 
conclusion by subordinating the content of the premisses to 
them? Aristotle was therefore right in his general idea 
of the superiority of the first figure ; we may also share the 
interest which he took in justifying the other figures by 
these changes of form ; but it is true that in practice it is 
seldom of much use to carry them out ; in considering the 
fourth figure just now we seemed to find such a case ; the 
inferences of the second and third figures are too transparent 
to need this assistance. 

92. It is therefore sufficient to mention that in the names 
of the moods of the two last figures the scholastic logic has 
indicated the operations necessary for this purpose by the 
letters m s p c. Thus m implies the transposition (meta- 
thesis) of the premisses : s and p tell us to convert, purely 
(simplidter) or impurely (per accident)^ the proposition whose 
characteristic vowel they follow : the meaning of c y reduction 
to impossibility (per impossibile duetto), is the only one 
which is not quite so simple, and may be at once illustrated 
by the case of Baroco. The premisses here are, ' all P are 
MJ ' some S are not MJ and the conclusion, ' some S are 
not P.' If we suppose this conclusion to be false, it follows 
ad contradictoriam that 'all S are P.* If this were so, 
and if this new minor premiss, 'all S are P, 9 were sub- 
ordinated to the given major, all P are M] it would follow 
in Barbara of the first figure that 'all S are M. 9 But this 
result contradicts the given minor 'some S are not M 9 } it 
was therefore wrong to deny the truth of the conclusion in 
BarocO) and that conclusion, * some -S 1 are not P 9 is right. 
The other operations scarcely need illustrating. We have 
lately seen how, by transposition, #/, of the premisses, and 
impure conversion, /, of the conclusion, which was then 


drawn in the first figure, Bamalip of the fourth is reduced 
to the first. Camestres of the second, ' all P are M] ' no S 
is M} ' no S is Pj gets by transposition, m, of the premisses 
and pure conversion, s y of the minor, the new premisses 
* no M is S,' ' all P are M 9 * from which it follows in Celarent 
of the first figure, 'no P is '; this conclusion further 
requires pure conversion, s, in order to yield ' no S is P,' as 
required by Camestres. Darapti of the third figure runs, 'all 
J/are PJ 'all J^f are S,' 'some Sare />'; the impure con- 
version, /, of the minor gives the premisses 'all Mare P* 
'some S are MJ and the resulting conclusion in Darii of 
the first figure, 'some 6* are P,' requires no further transforma- 
tion, being immediately identical with that of Darapti. 

93. Thus far we have conceived of the premisses as 
categorical judgments of the form ' S is P.' But the course 
of our thoughts may also suggest them in an hypothetical 
or disjunctive form. These differences, important as they 
are for the judgments as such, are not so for the formal 
connexion of the syllogism ; they always belong to its 
content, and it is only necessary to take note of them, not 
to alter the ordinary syllogistic rules on their account. 
This is most obvious where we have two hypothetical 
premisses, in each of which two of the three propositions 
MSP are connected as protasis and apodosis. Just as 
with categorical premisses where MSP denote three 
concepts^ the inference in Darii is as follows : ' P is always 
true if M is true, M is sometimes true if S is true, there- 
fore Pis sometimes true if *S T is true'; in Camestres, 'Mis 
always true if P is true, M is never true if S is true, there- 
fore Pis never true if S is true'; in Disamis, ' M is some- 
times true if P is true, Mis always true if S is true, therefore 
P is sometimes true if S is true/ 

The cases are more peculiar when the major premiss is 
hypothetical and connects universally a consequence F, 
expressed in the apodosis, with a condition G, contained in 
the protasis, while the minor is categorical and affirms or 


denies either G or F of all or some instances of S. The 
simplest way is to class these cases with the immediate 
inferences from judgments, for condition and consequence 
are related as subalternans to subalternata. Firstly, then, 
the fact that the condition G* is not true in certain cases of 
5 does not justify us in inferring ad subalternatam that the 
consequence F is not true in the same cases, for the same 
consequence may arise from other and equivalent conditions. 
But if the condition is true, we infer the truth of the con- 
sequence. This gives rise to two syllogisms, since G may 
imply either that F is true or that it is not true; (i) ' If G 
is true F is always true, G is true in all or some cases of S, 
therefore F is true in all or some cases of '; this is a 
modus ponendo ponens^ which posits the consequence by 
positing the condition, and it evidently answers to the 
moods Barbara and Darii in the first figure : (2) * If G is 
true F is never true, G is true in all or some cases of S, 
therefore ./MS not true in all or some cases of S'; a modus 
ponendo tollens, in so far as it does away with the consequence 
Fby positing the condition of its opposite, and obviously a 
counterpart of Celarent and Ferio in the first figure. 

In the opposite direction, ad subalternantem, the truth of 
the proposition F in certain cases of S does not prove the 
truth of the particular condition G on which it was found to 
depend in other cases, for the same consequence F may 
arise from several equivalent conditions. But the fact that 
F is not true in certain cases of S does prove that all 
conditions upon which it could depend, and therefore the 
particular condition G, are not true. The following syllo- 
gisms are therefore admissible : (3) * If G is true F is always 
true, F is not true in all or some cases of S, therefore in all 
or some cases of S G is not true/ a modus tollendo tollens^ 
which by doing away with the consequence does away with 
the condition which, had it been true, would inevitably have 
given rise to it; it corresponds clearly to Camestres and 
Baroco of the second figure : (4) 'If G is true Fis never 


true, F is true in all or some cases of 6", therefore in all or 
some cases of G is not true,' a modus ponendo tollens^ 
which by positing a consequence denies the condition 
under which it would have been impossible ; it repeats 
Cesare and Festino of the second figure. Lastly, we may 
reflect that the fact that G is not true may also imply that 
F is or is not true, in which case we get the syllogisms, 
(5) ' If G is not true F also is not ever true, in all or some 
cases of G is not true, therefore in the same cases Fis not 
true/ a modus tollendo tollens without any peculiarity, merely 
translating the ponendo ponens into the negative : (6) ' If G 
is not true F is always true, in all or some cases of 6* F is 
not true, therefore in these cases G is true,' a modus tollendo 
ponens, which was wanted to complete the possible com- 
binations of condition and consequence, positive and 
negative ; it posits the truth of a condition by doing away 
with the consequence which would necessarily follow if it 
were not true. An easy change in the form of expression 
shows that these two last cases also belong to the second 
figure ; the latter of them might be put thus, ( If non-G 
is true F is always true, F is always or sometimes not true, 
therefore non-G is always or sometimes not true.' As this 
exhausts everything that can be proved from the relation of 
subalternation, there are no consequences of this kind which 
could be classed under the third figure. 

94. These syllogistic devices are in my mind of less im- 
portance than a circumstance which I never find thoroughly 
considered in connexion with the present subject, the 
circumstance that all these inferences refer merely to a 
relation between the condition G and its consequence F, not 
to that of a cause G to its effect F. It is only in the world 
of thought that a condition G, if it is once suppose4 to be 
true, always has the consequence which by a necessity of 
thought belongs to it ; in the real world the cause G, even 
if it exists and is operative, may always have its effect F 
frustrated by an opposing force U. It being transferred to 


actual events, therefore, all these inferences require to be 
modified in ways which applied logic will show us : thus it 
is not allowable to conclude that wherever the cause G 
operates its effect F is necessarily a fact, nor to assert that, 
if is a cause of hindrance to F> where this hindrance 
exists F cannot exist ; G also in its turn may be hindered 
by a 7, or F may be realised in spite of it by a third cause F. 
In pure logic, therefore, it is quite an improper description 
of the cases which we have been dealing with to say, that 
their minor premiss expresses the real existence of G or F; 
the truth is that these two simple letters stand here for 
judgments of the form 'S is P'\ it is only the logical 
admissibility or necessity of this connexion of thought 
between 6" and P which the minor premiss asserts in regard 
to certain cases of S, while the major connects it with 
another similar relation between S and <2, so as to form an 
hypothetical judgment of universal validity. I will not 
pursue this point further here ; I have made my exposition 
somewhat prolix in expression with the view of indicating 
how the matter really stands. 

05. If it is true of a subject Z that it is either P, Q, or R^ 
or that it is both P, Q, and JR, or that it is neither P, Q, nor 
J, we first substitute for this triple predicate the simple U, 
and call 27 in the first case disjunctive, in the second positive, 
in the third negative. If anyone takes the not absolutely 
necessary trouble to follow the application of such disjunc- 
tive, copulative, and remotive premisses in the syllogism, 
he will find these results, (i) If the major premiss is Z U, 
and in the minor S Z an S is subordinated to Z, the 
ordinary conclusions S U of the first figure follow, and U 
has in them the same meaning as in the major : (2) If the 
universal major is Z 7, the minor c7, and U is in one of 
them positive or disjunctive, in the other negative, we get 
the negative conclusions S Z of the second figure with the 
quantity of the minor : (3) from the major U Z with a 
positive or negative 7, and the minor US with a 7 of the 


same or the opposite quality, there result the conclusions 
Z, always particular, of the third figure : (4) in the two 
latter cases, where U having become the middle term 
disappears from the conclusion, its multiplicity is entirely 
without significance ; what follows follows all the same if the 
position of one only of its members in the two premisses be 
taken into account. The result is equally little affected if the 
universal major Z U has a minor which affirms or denies of 
the particular subject Z one of the members of U. If the 
major distinguishes only two alternatives and says, * all Z 
are either P or <2,' and the minor ' this Z is P ' or ' this Z 
is not P, 9 it follows that ' this Z is not Q ' or this Z is Q: 
These consequences explain themselves from the nature of 
contradictory opposition ; they can be reduced, but without 
any conceivable advantage, to the first figure ; * every Z 
which is not P is <2, this Z is a Z which is not P, therefore 
this Z is a Q. 9 The same unfruitful reflexions may be 
extended to a U of more than one member in the major 
premiss, for we can always make any number that we 
choose of its members into the subject, and say (with only 
a bipartite 7), ' every Z which is not P and is not Q is 
either J? or T.' Lastly, polylemmas (dilemmas, trilemmas) 
are syllogisms with a disjunctive U of many members in the 
major Z 7, and the same number of minors, which taken 
together affirm of each one of the members of U the same 
further consequence T. These are not cases of new logical 
forms but only new applications of old ones, and we may 
return to them in our applied logic. 

96. On the other hand, I have no intention whatever of 
coming back to the doctrine of chains of inference. Every 
conclusion of a syllogism may conceivably become the 
major premiss of another syllogism : the first is then called 
the prosyllogism of the second, and each one that follows 
the episyllogism of the one which preceded it. A mere 
comparison of the names of the moods shows us at once 
many properties of the chain thus produced. If its last 


member is to be universal, the whole series of prosyllogisms, 
and therefore the whole chain, must be in the first two 
figures ; the entrance of any member in the third figure 
produces a particular conclusion, which never leads back 
again to universal conclusions. If one of the syllogisms has 
a negative conclusion, the conclusions of all episyllogisms 
are negative ; and a chain can only end with a conclusion 
at once positive and universal if it is in Barbara through its 
whole course. It is moreover usual to require, on the 
analogy of the simple syllogism, that the major premiss of 
the first prosyllogism should furnish the predicate P of the 
ultimate conclusion, and the minor of the last episyllogism 
its subject S: it would only need patience to find the rules 
for the formation of such a series, but I cannot see of what 
use they would be. If the conclusion of a prosyllogism, 
which is also the major 1 premiss of the episyllogism, is not 
expressed, the series give rise to the two forms of Sorites. 
The Aristotelian form, ' A is B, B is C, C is Z>, therefore A 
is DJ includes each concept in the one which follows ; it 
thus proceeds from the lower to the higher, and is produced 
by suppressing the conclusions, which we could elicit from 
each pair of members as follows, ' B is C, A is B, therefore 
A is Q and then, ' C is D, A is C, therefore A is D? 
The other form, a late discovery of Professor Goklenius of 
Marburg (1547-1628) takes the opposite direction ; its 
premisses, *B is A, C is B> D is C . . ./ suppress the con- 
clusion of the two first members, ' C is A? which as major 
premiss to the third gives the conclusion of the chain in the 
first figure, ( Z> is A.' 

A. Syllogistic Inference. Inference by Subsumption. 
Inference by Induction. Inference by Analogy. 

97. The logical truths of which the mind had gradually 
become conscious in dealing with its ideas were provision- 

1 [' Minor'' premiss, in the Aristotelian Sorites. The author's words 
only apply to the Goklenian form.] 

Chap. Ill,] SUBSUMPTION. 129 

ally summed up by the disjunctive judgment as follows : 
every S, which is a specific form of J/, possesses as its 
predicate a particular modification of each of the universal 
predicates of M to the exclusion of the rest. The problem 
which remained was to discover the intellectual operations 
by which this required specific mark could be determined 
for a given S. This problem is not solved by the Aristo- 
telian syllogisms ; they confine themselves to placing the 
subject of their conclusion in relation merely with the 
universal form of the predicate mentioned in the major 
premiss ; so that in spite of the manifold development given 
to them and their possible varieties by the acuteness of 
earlier logicians, they are merely the expression, formally 
expanded and completed, of the logical truth already 
embodied in the disjunctive judgment. Like the im- 
personal judgment, which, by distinguishing subject and 
predicate, made formally explicit a division already indi- 
cated in the concept, without telling us anything new about 
the mutual relation of the members thus produced, the 
Aristotelian syllogism in its first and most perfect figure, to 
which we mentally refer the others, merely distinguishes in 
two separate premisses the universal rule and its particular 
application, which were already similarly related in the 
disjunctive judgment. Thus the Aristotelian syllogisms, 
constructed as they all are on the principle of placing one 
concept within the circuit of another without further defin- 
ing its position, may be included, under the general name 
of inference by subsumption^ and considered as the first and 
most elementary form of the new group of intellectual 
operations. We will now attempt to show what is the nexj: 
step in advance which they compel us to take. 

98. As the most graphic illustration of the idea upon 
which inference by subsumption is based I choose the 
mood Darii 1 , which expressly brings a particular case in 
the minor premiss under the universal law contained in the 

1 {Sic. According to ordinary rules the example is in Barbara^ 


major. ' All men are mortal,' says this mood, ' and Caius is 
a man/ whence it concludes, * Therefore Caius is mortal/ 
clearly meaning that by this conclusion a truth which was 
not established before is now made certain by the truth of 
the two premisses and their relation to one another. But 
as early as the scepticism of antiquity the objection was 
made, that it is not the premisses which guarantee the truth 
of the conclusion, but that the conclusion must already hold 
good in order that the premisses may do so. Where, in- 
deed, would be the truth of the major premiss, 'all men are 
mortal,' if it were not already certain that Caius participates 
in this property? And where would be the truth of the 
minor premiss, * Caius is a man/ if it were still doubtful 
whether among the other properties of humanity he had that 
of mortality also, which the major itself alleges as a universal 
mark of every man ? Instead then of proving the truth of 
the conclusion by their own independent truth, the two 
premisses themselves are only true on the supposition of its 
truth, and this double circle seems at first to make the 
syllogism logically quite inoperative. 

99. The weight of this objection is not to be got rid of 
by denying it : we will follow out its applications in various 
cases. If we suppose the major premiss M P to be an 
analytical judgment, if, that is, we assume P to be a fixed 
mark without which the content of M cannot be cojnpletely 
conceived, then certainly the universal validity of the major 
is independently established; but then the minor cannot 
subordinate an S to M without already attributing to it this 
indispensable P, that is, without presupposing the conclusion 
in which that attribution ought first to find expression. If 
for instance we reckon weight in the concept of body, we 
form the majgr premiss, * all bodies have weight/ without 
fear of contradiction ; but we cannot go on in the minor to 
call air a body without involving the thought that air too is 
heavy, which we are not supposed to know until the con-, 
elusion. In general terms, the principle of subsumption 


requires that the subordinated individual should share the 
marks of its universal; but, conversely, nothing can be 
subordinated to a universal without already having the 
marks which the universal prescribes to it. 

The case would be different if we supposed the major 
premiss M P to be a universal synthetical judgment. Then 
the content of M could be fully conceived without involving 
the conception of P, though at the same time we should be 
certain, on whatever grounds, that P is always combined 
with M. The minor premiss would then merely have to 
show in *S the marks which make it an M, and then, and 
not till then, the conclusion would add the P which belongs 
to S in virtue of its subordination to M, but which had not 
before been part of the conception. In the practical 
employment of subsumptive syllogisms these assumptions 
are always made. When we assert, ' all men are mortal/ we 
conceive the physiological character of man to be fully 
determined by the rest of his known organisation, and 
regard mortality as a mark which need not be explicitly 
thought of when we mentally characterise him, because it 
follows inevitably from the organisation which determines 
our conception. And thus in the case of Caius it is enough 
to establish in the minor premiss the fact that he has this 
essential organisation, in order in the conclusion to ascribe 
to him <ts inevitable consequence. This is still more clear 
if we conceive the major premiss as hypothetical, and think 
of P as not a fixed and permanent but a fluctuating mark of 
M y a consequence which follows upon M under a certain 
condition x, a mark which under this condition M assumes 
or loses, a state into which it falls, or an effect which it 
produces. Then we have merely to subordinate S to M in 
the minor premiss in order to conclude that S also, if the 
same condition x operates, must exhibit the mark P. And 
as a matter of fact this is the form to which most of the 
effectual applications of the syllogism in science are re- 
cjucible ; they almost all show that S, being a species 

K 2 

1 3 2 f THE TffEOR Y OF INFERENCE. [Book I. 

will develop or experience under the condition x the same 
general effect P as we know in M. But as before with the 
analytical major premiss the question arose, with what right 
the minor could be asserted, so here with a synthetical 
major the question arises, with what right we can affirm the 
universal validity of this major itself. Mortality is to be a 
new mark, necessarily accruing to the organisation of man : 
but this universality can only subsist on the assumption that 
the conclusion is true, and it falls to the ground if some 
capricious Caius is found who does not die. It is clear 
what the answer to this will be : * of course/ it will be said, 
'every universal major premiss is false if there is a single 
instance in which it is not confirmed, and there is always 
this danger when the universal in question has been formed 
only by an unjustifiable generalisation from a number of 
observed instances : but where the necessary connexion 
of M and P is inherently demonstrable, the very universality 
of its truth provides against the contingency of a single 
capricious instance which might contradict it. In the 
example before us the matter is doubtful : to the ordinary 
mind the universal mortality of man is only an assumption 
based upon the overwhelming impression of countless in- 
stances, to which as yet no contradictory instance has been 
found : to the physiologist, as a consequence of the known 
human organisation, it is certainly a matter of settled con- 
viction, but not one which can be proved with the exactness 
he would wish. But in other cases the universal validity of 
the synthetical major premiss is guaranteed either by an 
immediate perception, or by proofs which reduce a given 
matter to such a perception, and in these cases the syllo- 
gism suffices for securing a particular piece of new know- 
ledge ; for all that this requires is perfectly practicable, viz. 
the subordination of an 6 1 to an M, which here really fulfils 
the function of a middle term in connecting S with a pre- 
viously unconnected PJ 
100. I leave it for the present an open question whether, 

Chap, in.] INDUCTION. 133 

and how far, the immediate perception of the universal 
truth of a synthetical judgment is possible ; for so much is 
at once clear, that in any case we shall be only very rarely 
in a position to rest the content of a universal major premiss 
upon this ground ; countless universal judgments are ex- 
pressed and used for inferences, without the possibility of 
either themselves passing for immediate perceptions or 
being reduced to such by any practicable method of proof. 
This wide field of intellectual activity cannot be simply set 
aside as invalid, nor can it subsist without logical rules of its 
validity. These rules we have to look for, and there are two 
which we want. For the effective use of the syllogism it is, 
firstly, necessary that we should learn to find universal major 
premisses, based neither on an immediate certitude nor 
upon the antecedent experience of their truth in every 
single instance ; it must be possible to assert the universal 
mortality of men, both before it is understood as the neces- 
sary consequence of certain conditions, and also before we 
have tested every individual man to see whether he is 
mortal. A second rule is necessitated by the minor pre- 
miss. There are many cases in which we are able to 
subordinate an *$" to M because we have found in S all the 
marks which ^/"prescribes to its several species, but in most 
cases this is impracticable ; even in the case of the Caius of 
our minor premiss no one will consider it necessary or 
possible, that in order to acquire the right to put him in the 
genus man we should test all the properties of his organisa- 
tion. If then the really fruitful exercise of thought is to be 
possible, there must be a method for finding minor pre- 
misses which subordinate a given subject to a genus before 
it has been shown to possess fully all the marks of that 
genus. The two methods which I am here requiring admit 
(though this is not of essential importance), of being at- 
tached to somewhat modified forms of the second and third 
Aristotelian figures. 

101, The problem of all inferential processes is naturally 


this, from given data or premisses to develop as much new 
truth as possible ; how this is done, is in itself quite imma- 
terial ; the method will be determined by the form of the 
premisses, and these we have to take as experience, internal 
or external, offers them. Now it often happens that the 
same predicate occurs or does not occur, not only in two, 
but in very many different subjects P, S, TJ F, W, and the 
question is, what consequence can be drawn from the pre- 
misses, PM 9 SM y TM 9 VM, , . . , which belong in form to 
the second figure of Aristotle. It is clear that in their mul- 
tiplicity they do not suggest an inference which would 
connect together any particular two of their subjects ; so 
far as we aim at such an inference, we can only effect it by 
confining ourselves with Aristotle to two premisses and 
observing the rules of the second figure. But it is equally 
open to us to try whether this recurrence of M in such 
different subjects tells us anything about the significance of 
M itself, which accordingly would not disappear in the con- 
clusion. Such an experiment is what the natural mind 
infallibly makes when experience furnishes such premisses, 
and it is guided in its experiment by the universal principle 
which dominates all its activity, the principle of translating 
a given coexistence of ideas into a coherence between their 
contents. Where we observe the same mark in different 
subjects, we are predisposed to think that the agreement is 
not a chance one, and that the different subjects have not 
therefore stumbled upon the same predicate each through a 
special circumstance of its own, but are all radically of one 
common essence, of which their possession of the same 
mark is the consequence. P 9 6", T 9 V will accordingly be 
different, but still co-ordinate as species under a higher 
concept 2 ; it is not as different individuals, but only as 
species of the genus S, that they bear the common mark M 
as a necessary mark of that genus. Our conclusion there- 
fore runs as follows, 'all S are M } ; and in this conclusion 
S stands for the higher universal to which we subordinate 


the individual subjects, and for the true subject of the M 
which before appeared as a common attribute of those 
individuals. Such a process of inference is the simplest 
case of Induction, and under this name forms our second 
member in the group of inferences based upon the sub- 
ordination of manifold elements to the unity of a uni- 

102. This process however seems only to solve imperfectly 
the problem which was set to it, that of producing universal 
major premisses for subsumptive syllogisms. For everybody 
agrees in objecting to induction, that if it is complete its 
information is certain but not new, while, so long as it is 
incomplete, it is new but not certain. If P, S, T, V are all 
the species of 2 which exist, and if each already has a 
premiss informing us that it is M 9 the conclusion can only 
sum up these premisses in a universal judgment, * All 2 are 
J/"'; but it cannot even logically be changed into the general 
judgment, * Every 2 as such is M'\ on the contrary, it 
remains quite uncertain whether the species of 2 merely 
participate as a fact in the common M> and each ultimately 
for a special reason of its own, or whether the universal 
nature of 2 really contains the one and selfsame reason 
which makes M a necessity to all its species. If, on the 
other hand, besides those subjects which are combined with 
M in the premisses, there are other species of 2 of which 
those premisses say nothing, then the conclusion is an 
unjustified inference ad subalternantem from the truth of a 
limited number of instances to the truth of all, an inference 
which may have probability in various degrees, but never 
reaches certainty. 

It appears to me, however, that these observations, right 
as they are in themselves, confuse the pure meaning of a 
logical form with the difficulties of its effective application, 
and that there was the same error in the criticism made 
upon the value of the Aristotelian syllogism. The leading 
idea of that syllogism, that every individual derives its right 


and obligation to the possession of its predicates through 
dependence upon its universal, is without doubt logically a 
perfectly valid principle, and exhibits in its true light the 
internal construction of the content of thought in question. 
It does not lose this logical significance because the truth 
of the universal includes or, if we prefer it, presupposes its 
truth in all particular instances ; on the contrary, the very 
meaning of the syllogistic principle is that the two are 
inseparable. Whatever therefore may be the way by which 
in practice the mind has arrived at the truth of the pre- 
misses, when they are once found the first Aristotelian 
figure does express by its structure the inner connexion 
of the completed content of thought, though it probably 
does not at all express the division of intellectual labour 
by which we made it our own. Considered in this way the 
subsumptive syllogism is the logical ideal> to the form 
of which we ought to bring our knowledge, but it is not the 
general instrumental method by which we compose that 
knowledge out of the material given to us. 

I have a similar remark to make about induction; the 
logical idea upon which it rests is by no means merely 
probable, but certain and irrefragable. It consists in the 
conviction, based upon the principle of identity, that every 
determinate phenomenon M can depend upon only one 
determinate condition, and accordingly that, wher^ under 
apparently different circumstances or in different subjects 
P, $*, JJ 7 the same M occurs, there must inevitably be in 
them some common element 5, which is the true identical 
condition of M or the true subject of M. It would be quite 
unjustifiable to object, that as a matter of experience the 
same consequence M is often produced by different equiva- 
lent conditions, and the same predicate M may occur in 
extremely different subjects. Such an objection just shows 
the confusion, which we condemned above, between the 
logical rule and the conditions of its application. If there 
are two equivalent conditions for a result M, it is not in 

Chap. III.] ANALOGY. 137 

virtue of that which makes them different, P or S, but of 
that which is the ground of their equivalence, that they are 
really conditions of the same result : so long as we cannot 
separate this common characteristic in the two, we have not 
yet found the true 2 of the conclusion, and have not there- 
fore carried out the induction in the way in which it de- 
mands to be carried out. Again, if the same M is found 
as predicate in a number of extremely different subjects, 
and subjects (as is usually the case in practice) the several 
sums of whose marks are only partially known, we may 
of course make a great mistake if we combine what is 
common to the known marks of all of them, and then 
assume it to be 2, the true subject of the mark in question 
M. I do not deny that in the practice of induction we are 
often placed in such unfavourable circumstances ; but all 
these difficulties in carrying out the inductive principle do 
not alter its universal logical validity when it asserts, that 
wherever different conditions have the same result M, or 
different subjects the same predicate M, there must be 
discoverable one and only one quite determinate 2, forming 
the single invariable condition or the single true subject, to 
which the predicate or the result M is to be universally and 
necessarily ascribed in a conclusion of the form, ' every 2 is 
M. 1 We leave it to applied logic to observe the rules by 
which we may succeed in discovering this 2. 

103. I introduce the third form of this group under the 
somewhat arbitrary name of the inference of analogy. In the 
third Aristotelian figure, MP, M S, as in the second, the 
structure of both premisses being exactly the same, there is 
nothing in their position to lead us to distinguish major 
from minor, or to limit their number to two. On the con- 
trary, experience will often show us a larger number of them, 
M P, MS, MT, M If] will show us, in other words, that 
a number of different marks does or does not occur in the 
same subject. These data cannot be rejected by the mind, 
and it employs them to form an inference which is just like 


the one described above, only in the reverse direction. 
Here, as there, it is guided by the assumption that the 
different predicates have not united in the same subject 
M by a number of unconnected chances, but that they 
must be coherent and owe their coexistence to the 
presence of one condition \ they belong to M because 
M is a FT, and it is this sum of marks which in its com- 
pleteness constitutes the nature of IT ; and M 9 being a 
species of FT, has a right to unite them all in itself. Thus 
from these premisses we form the conclusion, ' M is a IT,' 
and have so executed our second task of finding for the 
subsumptive syllogism a minor premiss by which a concept 
M (there called S) is subordinated to another concept II 
(there called M). 

104. Yet this task, like the former one, seems to be but 
badly executed, for analogy, like induction, is liable to the 
charge that, if complete, it tells us nothing new, and if 
incomplete, nothing certain. If the premisses already give 
M the marks necessary to make it a II, we gain nothing in 
knowledge of fact by actually bringing it under this concept; 
the change is merely in the form of our apprehension of the 
given content. But in most cases the premisses give only 
a part of the predicates necessary to IT, and from the 
presence of these we conclude without certainty to that 
of the rest, by which alone the whole of II is realised in M. 
When we have to do with concrete objects, which in their 
totality consist of countless marks, in great part unknown 
to us, in part difficult to observe, this is always the case : 
from a few properties which we actually observe in an 
object, we conclude that it is a metal, an animal of a 
certain kind, an instrument for a certain purpose. It is 
needless to say that numerous mistakes in the employment 
of analogy arise from this fact ; but here also the difficulty 
of the application does not diminish the value of the logical 
principle. That principle asserts, that no rightly conceived 
content of thought consists of an unconnected heap of 


marks, which we may increase at pleasure by adding no 
matter what new elements; what other marks as yet un- 
observed can combine with tne observed marks and what 
cannot, is already decided, not indeed by one mark, but by 
a given combination of several, in which each is determined 
by all the rest ; this is why we are able from the incipient 
form of M given us by the premisses to infer its further 
completion and continuance ; there is always therefore one 
and only one IT, which makes legitimate and possible the 
union of marks given in AT, and at the same time the 
addition of others not given. This ideal of thought, which 
in itself is quite true, only requires, like every form of 
thought, to be realised in suitable, not unsuitable, matter. 
It is not any casual pair of predicates in an M which suffice 
for inferring the rest ; many such combinations may belong 
to some other concept II 1 or II 2 as well as to II ; in contrast 
with such unessential marks we shall require essential ones 
in the premisses, a requirement which is always made in 
practice, and which it is left to special knowledge of the 
matter in question to meet. The most important source of 
inexactness, however, is that all the forms of inference 
hitherto mentioned give the predicates only a universal 
form, without indicating their measure, specific modifica- 
tions, and mutual determination. So long as the premisses 
only say, c M is heavy,' { M is yellow/ * M is liquefiable,' 
etc., we certainly find in such data no decisive ground for 
pronouncing M either to be gold or to be sulphur : but 
this is just why such premisses are only met with in abstract 
logic; in actual practice attention is always given also to 
the particular amount, nuance, and combination of the 
predicates, and from this incipient characterisation its con- 
tinuity with the completed IT is inferred. It is just this 
universal practice of the natural mind for which we have 
to find a theoretical basis in new logical rules, and these we 
must now go on to consider. 


B. Mathematical inferences. Inference by substitution. 
Inference by proportion. Constitutive equation. 

105. I will put together once more, and from different 
points of view, the motives which impel us to go beyond 
the syllogisms and look for new forms of thought, and for 
this purpose I will first touch upon the nature of the judg- 
ments which the ordinary theory conceives of as members 
of the syllogism. In judgments of the form * is PJ as 
I have already observed, language expresses the predicate 
with a universality with which it does not belong to its real 
subject, and logic usually concedes this when it asserts that 
not only does the predicate contribute to the determination 
of the subject, but the subject also to that of the predicate. 
When we say, ' this rose is red,' we do not mean that it has 
a general indefinite red, or any casual shade of colour which 
happens to be included under the name red ; it is rose-red 
only that we always have in our mind, or, more accurately 
still, the precise red of this rose. If then we wished to 
express our thought exactly, we should have to say, 'this 
rose is red with the redness of this rose. 7 In this apparently 
quite barren proposition the logical activity would show 
itself in the fact, that the perceived property of the rose is 
no longer apprehended as an isolated thing, without any 
other home in the world ; in regarding it as a kind of red in 
general, which occurs elsewhere and holds good indepen- 
dently of this instance, the mind, as we said before 1 , 
objectifies its perception; it gives to what is perceived a 
definite place in the world, which makes it something on its 
own account, and not a merely subjective excitation of the 
percipient at the moment. In this lies the logical gain 
which always results when the particular content of a 
perception is replaced in the judgment by the universal of 
which it is an instance. But at the same time of course 
there will be a logical loss, if we get no further than the 
i [Above, 3.] 

Chap. III.] QUANTITY. 1*41 

expression of this Universal, and if the other part of the 
perception does not get its due by addition of the parti- 
cularisation which is necessary to make the universal named 
equivalent to the individual intended. This loss is sustained 
by all ordinary judgments of the form just mentioned, and 
the Aristotelian syllogisms too confine themselves to dealing 
with the universal MOT the universal P. 

108. In this way they leave unsolved the particular 
problem which the disjunctive judgment suggested, and 
fail generally to satisfy the practical needs of thought as a 
living process. For already in the disjunctive judgment it 
was asserted, that it is not the universal predicate of its 
genus which belongs to the individual, but a definite 
modification of it, /, to the exclusion of the rest. What 
this p is, ought to have been made out by the syllogism ; 
and it could only have done so by supplying to the major 
premiss, which connects the genus with the universal P, a 
minor bringing out the peculiarity in virtue of which S is 
this particular species of the genus and no other, and must 
therefore have for predicate this and no other modification 
of P. This has not been done; the minor premiss also 
only mentioned generally the subordination of the indivi- 
dual to the genus, but not its specific difference from other 
species of it; hence the conclusion could only say what 
belongs ^.to the individual as a species of its genus, not as 
this species. It hardly needs to be further explained that 
this falls short of what the actual processes of thinking 
demand. If we argue, ' heat expands all bodies, iron is a 
body, therefore heat expands iron,' or, * all men are mortal, 
Caius is a man, therefore Caius is mortal,' everyone will 
feel the barrenness of this procedure, and will reply, ' Un- 
doubtedly heat expands all bodies, but each body in a 
different degree ; undoubtedly all men die, but the liability 
to die in one man is different from that in another; what 
we want to know for technical purposes or for administering 
a life-insurance company is, how iron expands in distinction 


from lead, or how the mortality of Cams is to be estimated 
in distinction from that of other men.' This then is what 
the new forms have to do ; they have to make the individual 
felt as a definite species of the universal, and so enable us 
to argue from its distinctive difference to its distinctive 

107. From another point of view we may notice the fact, 
that in logic it has been too exclusively the custom to use 
categorical judgments as illustrations, and therefore also to 
represent the inclusion of one concept within another as the 
most frequent and most important of logical operations. 
In the living exercise of thought this is by no means the 
case ; we are seldom concerned in practice to determine a 
mark which belongs to a concept once for all, or in the 
circuit of which the concept is to be classed; most fre- 
quently we want to know what variable mark P will occur in 
a concept 6* if S is subjected to the condition x. Questions 
of this kind are being raised at every moment by life,' 
science, and art. We must admit that the ordinary syl- 
logistic method does not entirely overlook such cases ; but 
it is only an imperfect way of dealing with them to make 
P the universal result of the coexistence of x with M in a 
major premiss, and then to ascribe P to S, again only 
universally, by subordinating to M or MX. What good 
is it to say, c if a man is offended he gets angry, Caius is a 
man, therefore if he is offended he will get angry'? What 
we want to know is, how Caius, being the person he is, will 
get angry, and consequently how far we may go with him. 
The subordination of Caius to the concept of humanity 
helps but little to answer this question ; we must look for 
the special characteristics which distinguish Caius from 
other persons, and must then have the means of calculating 
the effect which offence will have upon these characteristics. 
This may be briefly expressed thus : our inferences cannot 
be derived from extensive relations between the given 
concepts, but only from their content ; without making the 


unprofitable circuit through the universal genus, we have 
to determine directly from the given marks of a subject, 
and from the accruing condition ,#, what new marks will 
show themselves or what changes will take place in the old 

108. Considered from this point of view, the new forms 
which we have to look for group themselves with the 
inferences from analogy. For these also concluded from 
the presence, absence, and combination of certain marks 
in an S the necessary presence, absence, and mode of 
attachment of other marks in the same subject. We may 
doubt indeed whether such inferences from content to 
content, from mark to mark, are possible on merely logical 
grounds, and whether the few which really are possible 
are not already anticipated by the familiar logical doctrines 
of the compatibility of disparate predicates, the incom- 
patibility of contraries, and the necessary choice between 
contradictories : statements such as, ' where / is there q 
must be/ will after all (it may be said) be supplied by 
experience alone, with the single exception, with which 
we wish to have no more to do here, when q is already 
included in the content of / or / in the extent of q. In 
itself this doubt is right ; all assertions about the necessary 
connexion or incompatibility of two predicates, with the 
exception of the cases last mentioned, can never be based 
upon any evidence but that of observation ; but it is still 
a question whether logic, with the means hitherto at its 
disposal, has made even these necessarily presupposed 
facts yield all the consequences which they might be made 
to yield : that it has not done so, I can show more shortly 
by exhibiting the actual forms of inference to which I refer : 
in the natural use of the mind they are current and familiar, 
and all that is done here is to give them the place which 
belongs to them in a system of logic. 

109. Let us leave to the major premiss of our new figure 
the form, 'all M are P' or M~P \ to the minor however 


we will give, not the indefinite form, ' S is an M in general/ 
but the definite one S=sM; that is, S is that species 
of M which we get if we conceive the whole structure of 
the marks in J/as determined or modified by the influence 
of a specific condition s. The conclusion will then have 
to be, * S is cr PJ and it will assert that S, so far as it is 
this distinctive species of M characterised by s, possesses, 
not the universal mark P, but that specific impression 
of it, cr P, which the influence of s must produce in the 
structure of M. To avoid misunderstanding, it should 
be observed that the influence of a condition s upon the 
whole structure of M may transform the different marks 
of M in extremely different ways; each one of these 
transformations is a result of s, and on that account I 
have employed the kindred letter cr in cr P : on the other 
hand it is not generally right, though it may be so in 
particular cases, to make the modification of a mark equi- 
valent to the modifying condition ; therefore the conclusion 
here could not be indicated by sP. In the form however 
which we have given to the conclusion, it would be merely 
the indication, not the solution, of a problem. What is 
wanted is to give a name to this <r P, and to show how 
P is changed by the influence of s upon M. This remains 
impracticable so long as we produce M merely in this 
simple form of a universal concept provided with a name : 
in order to know how s influences M, we must analyse 
the content of M into its several parts, and observe in 
what manner they combine. Nobody, for instance, will 
undertake to judge how the working of a machine will 
change under the influence of a force s, so long as he 
merely has the machine before his eyes as a simple object 
of perception, M, a steam-engine in general ; he must first 
get to know the inner structure, the connexion of the 
parts, the position of a possible point of action for the 
force s, and the reaction of its initial effect upon the parts 
contiguous to. that point. Accordingly, it is only by sub- 

Chap. Hi.] THE SYMBOLS. I^g 

stituting for the condensed expression or concept M the 
developed sum of all its constituent parts, with attention 
to their mutual determinations, that we can hope to follow 
the influence of s, and so determine, firstly, what is the 
whole nature of S which =.f J/J and, as a consequence, 
what is the modification <r P of the predicate P which 
belongs to this S. As a matter of fact, this second part 
of the problem is always included in the first ; the specific 
modification of a particular predicate for S cannot possibly 
be found without first finding the total change produced 
in M by s, on which the modification depends; for if 
P were part of a different concept JV, the effect on it 
of the same condition s would not be the same as when 
it is a part of M. For this reason I shall take no more 
notice of the inference to <r P, but shall consider the 
problem of the new form to be to determine ^ M, and 
give it therefore the form, 

Major premiss : M=afrx<:x z . . . 

Minor premiss : S =sM. 

Conclusion : S =s(a bx ex* . . .) 

from which, in regard to single predicates, e.g. ^, there 
would follow the definite conclusion, ' S is s . bxj instead 
of the indefinite one, '-5* is bx? 

110. There is always a danger in expressing very different 
and yet connected cases by the simplest possible symbols ; 
to avoid misunderstanding, therefore, I add the following 
observations. By a, &, t, x I wish to be understood, 
speaking generally, different marks of a concept Jlf 9 which, 
when completely enumerated, constitute the whole of M. 
But in each different concept these marks stand in the 
most different kinds of relation to one another, and these 
relations are not expressed in my formula ; the double 
sign + has been employed as a faint indication of their 
possible variety. These signs, -f and , do not suffice for 
a full expression even in a case where M does not mean a 



conceptual content of qualitatively different marks, but a 
mere whole of quantity composed of the commensurable 
quantitative parts a, , c, x. The only symbol of a more 
exhaustive kind would be that of the mathematical function 
in general, which we used before, M~ F (a, b, c, x . .) ; 
but this would have the disadvantage of merely calling up 
to thought all modes of connexion between the parts, with- 
out giving a sensuous illustration of any. The form of the 
series a -f bx + ex* is also an arbitrary symbol the x only 
indicates a possible difference of value in the marks, one of 
which, x, leaves only one other, a, entirely free, while it 
accompanies the rest as a determining condition. The s of 
the minor premiss and conclusion appears here as a multi- 
plying factor; this is similarly intended to represent to 
sense, by the simplest and most familiar form in which one 
quantity can influence another, the countless different ways 
in which any concrete condition may act upon the manifold 
content of any given subject. If we express by a letter 
placed underneath on the right any kind of change produced 
by a condition in any kind of given matter, and represent 
jfcfas a function of a, , t, x (i.e. M=<f> (a, b y c, x) ), we 
should in general only be able to represent the conclusion 
by =& (a n b n c n x s \ not by S=$ (a,, b n c n #.) ; for it is 
obvious that the effect of s may not always be merely to 
change the single marks, retaining their general connexion 
(as expressed by the second formula), but also (as expressed 
by the first) to change this connexion itself; in fact, a con- 
dition may so transform the whole structure of a concept 
that in its new shape it has to be subsumed under a different 
concept M 1 or N instead of the previous M. The ad- 
mission which I have now to add makes it unnecessary 
for me to go further into this point. 

111. The advantage which we anticipate from this figure 
of syllogism by substitution, the first of this second group, 
depends ultimately upon our knowing what the several 
parts of the conclusion mean, i.e. what that value a, or bx 9 


is which arises from the influence of s upon the developed 
expression of M. This, however, if it is not to be learnt 
simply by experience, can only be arrived at by thought if 
all these mutually related parts are pure quantities, and the 
relations between them those of mathematical combination 
and separation. Thus the effect in use of our figure is con- 
fined to the region of mathematics, and primarily to the 
relations of pure quantities. Only the peculiar nature of 
numbers, each one of which has an expressible relation to 
every other, allows us to disclose the hidden content of Jlf y 
by substituting its quantitative parts, in such a way that the 
condition s can really operate upon it, and that by applying 
the various rules of calculation, by cancelling incompatible 
and compounding compatible elements, the change which s 
necessitates in M can be really carried out and the form of 
the new result exhibited. On the other hand, if we replace 
commensurable quantitative parts by incommensurably 
different marks of a concept, these advantages disappear 
again ; the content of M is only imperfectly disclosed by 
such a method of substitution ; for we have no rule here, as 
we have in the case of numbers, by which to measure the 
effect of a condition acting upon these heterogeneous ele- 
ments. It is true that even in such cases we apply the 
general idea of substitution : if we want to know how a con- 
dition s will act upon a thing, of which we have only the 
concept M which its natural history supplies, we analyse M 
into its marks ; but the calculation of the effect which s will 
have on each and all of them, is based merely upon more 
or less indefinite analogies, suggested by experience or some 
chance feeling of probability. 

112. The fact that the use of the syllogism by substitution 
is confined to mathematics, cannot hinder us from giving 
it a place in the systematic series of forms of thought. For 
in the first place we must not forget that calculation in any 
case belongs to the logical activities, and that it is only their 
practical separation in education which has concealed the 

L 2 


full claim of mathematics to a home in the universal realm 
of logic. But it is not only because they are indispensable 
to a part of the work of thought, that these forms have their 
place here ; even in those cases where their demands can- 
not be realised, they are still the ideals of our logical effort. 
For if they can be applied directly to none but quantitative 
relations, it is true on the other side that wherever we are 
quite unable to reduce the object of our investigation to 
those relations, our knowledge of it remains defective, and 
that no other logical form can then help us to the answer 
which a mathematical treatment of the question, if it were 
practicable, would give us. It is hardly necessary in our 
days to draw attention to the fact, that natural science owes 
its existence to mathematics ; in other fields also we have 
learnt to prize the important aid of quantitative statistics in 
discovering the laws which govern the combinations of 
society ; and even in sciences which from the nature of their 
objects are farthest removed from mathematics, we often 
feel very clearly the need of connecting them with quantita- 
tive ideas. Moral philosophy may decide that every crime 
is punishable, without needing a mathematical justification 
for the assertion ; but every punishment which has really to 
be inflicted must have a measure, and this must be regu- 
lated by the measure of badness in the criminal will which 
has to be punished. If only it were practicable, the penal 
law itself would draw conclusions in our figure of syllogism ; 
it would break up every crime by substitution into its 
several elements, and from s M, i.e. by calculating the 
particular values of the single elements of the crime in this 
instance, and so the particular value of the whole, it would 
deduce <r P, i.e. the kind and amount of punishment which 
the particular instance deserves. 

113. There are other things however besides pure mathe- 
matics, and science has certainly succeeded in establishing 
links of connexion, even between incommensurable pheno- 
mena or attributes, which allow us to infer from one to 

Chap. HI.] PROPORTION. 149 

another. For logic on its part the next problem must be, 
to look for the forms in which such inference is possible, 
and so to supplement the imperfection of the substitutive 
syllogism. It would partly seem indeed that science has 
only succeeded in thus bridging the incommensurable by 
doing away with the incommensurability, and showing 
that two facts, a and , which at first appear to our per- 
ception entirely different in quality, really depend upon 
quantitative differences between commensurable circum- 
stances : I may recall how physics has reduced the quali- 
tative differences of our sensations of colour, tone, and 
heat to merely mathematical differences in commensurable 
motions of commensurable elements. If however we look 
more closely at these cases, we find the fact to be, not that 
our sensations, a and , are reduced to motions, a, and /3, 
commensurable with one another and with the sensations, 
but merely that the occurrence of a or /3 and its effect upon 
us is represented as the condition upon which the sensation 
a or b necessarily arises. The perceived colour a remains 
just as incommensurable as ever with the vibration of 
ether, a, by which its origin is explained ; and if experience 
did not teach us that a is the consequence of , we should 
have no logical means of divining from a the nature of its 
cause a. What therefore science does in these cases is 
really to connect incommensurable elements in a way which 
allows us to conclude from one to the other. The original 
proposition that a and a, b and /3, do thus mutually point 
to one another, is due, as I said, to experience ; in deriving 
it from facts the laws of thought are doubtless applied, but 
there is no special form of thought involved such as could 
solve the insoluble problem of making commensurable what 
is really incommensurable. But when experience has in- 
formed us of the coherence of two such elements, a and a, 
then thought concludes that this coherence will be main- 
tained even in the event of their both changing, and that 
.therefore a definite change of a into a 1 must always be 


answered by one and only one definite change of a into a 1 . 
Again, these changes themselves, a a 1 and a a 1 , are not 
directly commensurable, either in kind or amount : if the 
number of vibrations of the sound-wave is increased by the 
amount 6= a 1 , i$ is true that a definite increase, daa\ 
in the heard tone depends upon it ; but this change in the 
pitch is a process quite different in kind from the increase . 
in the number of vibrations, and cannot be compared with 
it ; each quantity can still only be measured by a standard 
of its own, and their mutual coherence can be expressed 
as a fact and nothing more. But the changes in pitch are 
commensurable with one another, and so are the changes 
in the number of vibrations ; and if we refer these changes 
to d and b as their respective units, we may ask, By how 
many units m of the kind d does the pitch change, if the 
number of vibrations changes by p. units of the kind 8 ? 
m and ft then stand in a purely numerical relation. This 
relation may be infinitely various; but, as before, I shall 
not indicate the possible variety any further in the form 
which I give to this inference. I choose for its name and 
scheme the simplest form of proportion, E\e=.T\t y which, 
though it only illustrates the case in which mi^is a constant 
quantity, still sufficiently symbolises the logical idea implied 
in the process. 

114. I will illustrate that idea once more by a very 
elementary example. Two angles E and e are commen- 
surable; so are two segments of a circle 7*and /; but an 
angle and a segment are incommensurable and cannot be 
directly measured by any common standard : so too the 
difference of two angles, which again represents an angle, 
is incommensurable with the difference of two curves, 
which again forms a curve. Nevertheless, if it is once 
established that a certain length of curve t belongs to an 
angle e at the centre of a circle of a given diameter, and 
if we form the angle E by m times e and the corresponding 
curve Tby n times /, then the pure numbers m and n are 


commensurable, which tell us how many times the two 
intrinsically incommensurable units / and e have to be 
multiplied in order to find two corresponding members 
in the two series of angles and curves. For the circle 
geometry tells us that m n. Given therefore the two 
units, e and /, we only require to know a definite number 
E of e in order to arrive at the proper value of T by the 
proportion E\e~T\t. Expressed as a syllogism, then, 
the whole process would answer to the scheme, 

Major premiss : E : e = T\ /. 

Minor premiss : E = F (e). 

Conclusion: T=F(e.t 

115. I need hardly point out that upon this inference by 
proportion^ in the simple scheme of which I include all 
more complex relations between m and n, rests ultimately 
the whole possibility of bringing qualitatively different 
occurrences into such mutual dependence as allows us to 
calculate one from another. It is also scarcely necessary to 
observe that we can only expect this figure to be fully 
effective, so far as we succeed in reducing the relations 
of things to terms of pure quantity : we should justify this 
limitation in the same way as we did the similar limitation 
of the syllogism by substitution. In a more lax way we are 
constantly judging of things, even in ordinary life, on the 
ground of inexact proportions, which mostly pass into mere 
comparisons : a general likeness is found between the 
relation of a to b and that of a to /3, but the equal exponent 
of both is not precisely specified, and so the inferences 
drawn generally carry little conviction ; e.g. ' If one of these 
relations under a certain condition c has a certain result y, 
the other will have a generally similar result under the 
same condition/ 

I have only one more remark to add, in repetition of 
what I have already said, viz. that the form of proportion 


indicates a limit of knowledge. We find in it the inter- 
dependence of two members E and T merely expressed as 
a fact, and as such utilised for further purposes; on the 
other hand, the question remains unasked and unanswered, 
in what way, by what means, through what mechanism, so 
to say, the one member E sets about bringing the other T 
into any sort of dependence upon itself, especially into this 
particular sort. Of course there are a great many com- 
posite phenomena, in the case of which this question too 
can be answered : scientific investigation, as we said, has 
reduced many pairs of apparently disparate properties or 
occurrences to merely quantitative differences of commen- 
surable terms, and we are then able to see how it comes 
about that T must be connected with 2 9 and a particular 
increase of the one with a particular increase of the other. 
But there is a limit to this possibility: the ultimate dis- 
coverable laws of phenomena will always be found to 
involve determinate relations between disparate elements, 
which we can only accept as facts and utilise in the form 
of proportion, without being able to show the reason why 
the two elements must be proportionals. We refer many 
phenomena to the law of gravitation, the intensity of which 
is reversely as the square of the distance ; but hitherto at 
any rate no attempt has succeeded in showing how the 
distance contrives to weaken the force. We show how the 
sensible pitch increases with the increasing number of 
vibrations, and how our sensations in general, and in fact 
all our mental activities, change proportionally to physical 
motions in our organs ; and yet after all, tones and vibra- 
tions, mental functions and physical motions, remain for 
ever intrinsically incommensurable, nor do we ever ex- 
perience how the one contrive to compel the others to 
corresponding changes. From one disparate thing to 
another our thought has no means of transition ; all our 
explanation of the connexion of things goes no further back 
than to laws which admit of being expressed in the form of 


proportion; and these laws make no attempt to fuse the 
two elements into an undiscoverable third, but leave them 
both in their full difference, and merely point out that, in 
spite of their mutual impenetrability, they come as a fact 
under a common law by which they mutually determine 
one another. 

116. In the actual application of the inferences from 
proportions another defect, hitherto only briefly indicated, 
is tacitly supplemented by attending to an idea which 
necessarily accompanies them ; this supplementary idea we 
have now explicitly to recognise as having a place of its own 
in the systematic series of intellectual operations, the last 
place in the present group. In the above scheme the pro- 
portion between the changes of two marks E and T was 
represented as if it always subsisted between the two marks 
as such, it being indifferent in what subject they occur. 
Now there are, it is true, predicates which upon logical 
grounds, on account of their contrary or contradictory 
opposition, or because the one in any case includes the 
other, must be either present together or absent together in 
every subject : but there are no marks whose quantities and 
quantitative changes must always stand in the same pro- 
portion to one another, whatever be the nature of the subject 
in which they are united. On the contrary, it is just this 
nature which determines the exponent of their proportion ; 
and the same universally expressed marks E and T y which 
in one *S* can only coexist in the ratio n : m, are in another 
S l only possible in another ratio n l : m\ Heat expands all 
bodies, but the ratios of the degree of expansion to an equal 
increase of temperature are different in different bodies. 
In practice, where we always have to do with individual 
subjects, and have these in mind throughout, we do not 
need to state this limitation expressly; but logic is bound 
to emphasise the fact that only on the assumption of the 
limitation can we talk of using proportions. Nothing but 
the specific character of a given subject, in obedience 


to which all its marks mutually determine one another, jus- 
tifies us in concluding from a known value of one of 
them to the corresponding value of another according to 
a proportion which holds good for this subject only. This 
merely brings us back to the idea which lay at the root 
of analogy ; for it was only on the strength of the co- 
herence of all mutually determined marks in a concept, 
that we felt justified in inferring from a limited group of 
them to the necessary presence or absence of others, as 
we might infer from the beginning of a pattern to its 
continuance. This tacitly assumed condition must there- 
fore be added in order to complete the expression of the 
proportional syllogism, and its major premiss ought to 
stand thus, ( If is an M, for this S it is always true 
that T: t = E : e. y And the problem which logic pre- 
sents to us would not be merely to establish this major 
premiss through experience, in order then to bring a par- 
ticular case under it in the minor, l S is M ' ; it would 
rather be to show how a concept M can be found at 
all, such that the proportions required between every two 
of its marks can be derived from it. 

117. The means for the discovery of such an authori- 
tative or constitutive concept have already been indicated ; 
they lie in the fact that every mark is determined through- 
out by every other, though in very various way 4 s. The 
effect of this variety will be that, while in certain cases 
the presence of a single proportion between any two marks 
is sufficient to determine the rest, in others the know- 
ledge of certain essential marks is necessary in order to 
deduce the unessential from them, but knowledge of the 
unessential is not enough to establish with certainty the 
whole content of the concept. But I shall be clearer if 
I preface these reflexions by an instance of the actual 
realisation of our requirement in the shape of a very 
familiar and simple mathematical form of thought. Analy- 
tical geometry possesses in the equations^ by which it ex- 


presses the nature of a curve, just that constitutive concept 
of its object which we are looking for. A very small 
number of related elements, the indeterminate abscissae 
and ordinates in their combination with constant quan- 
tities, as constituting a primary proportion, contain, implicit 
in themselves and derivable from them, all relations which 
necessarily subsist between any parts of the -curve. From 
the law expressing the proportionality between the changes 
of the ordinates and the abscissae every other property 
of the curve can be developed, its course, its openness 
or closedness, the symmetricalness or unsymmetricalness of 
its parts, the uniformity or measure of alteration in its 
curvature at every point in it, the direction of its con- 
cavity or convexity, the area which it contains between 
any given limits. It is in view of these developments (the 
further course of which is too simple to need mentioning 
here) that we give the name of inference from constitutive 
equations to the method in question. The method itself 
is not confined to these geometrical problems; but the 
other and in some ways much more interesting examples 
supplied by other branches of mathematics, especially the 
calculation of variations, cannot be so easily represented 
with the simplicity requisite to symbolise the form of 
thought which we are considering. Natural science also 
could fyrnish approximations at any rate to what we are 
looking for. Chemistry would possess constitutive equa- 
tions for analogously compounded bodies, in which the 
different chemical elements take the place of co-ordinates 
and constants, if it could succeed in expressing by its 
formulae not only the quantitative proportions of the 
elements, but also, more exactly than its symbols at present 
do, the rule for the grouping of atoms and the general 
character of their interaction. 

118. Admitting the objection to the whole of this method, 
that like the preceding one it is not fully effective except in 
mathematics, we rebut in the same way as we did before 


the censure which it seeks to convey, and only examine 
it more in detail with a view of finding new ways to supple- 
ment what is still defective in the method. It is true that 
the apparent wealth of development from geometrical equa- 
tions is, from a logical point of view, more specious than 
real. In order to determine the form of the curve we give 
one of the co-ordinates x arbitrary values, calculate the 
corresponding values of y from the equation, and then con- 
nect the extremities of the perpendiculars (y) erected upon 
the extremities of the abscissae (x) so as to form a con- 
tinuous line ; the curve is therefore only the geometrical 
locus in which the countless results of a countlessly repeated 
proportion between different values of the co-ordinates 
are combined. As for all the new properties which we 
proceed to deduce, concavity, uniform or varying curvature, 
closedness or openness, falling or rising of the curve to this 
or that side, though at first they look like new marks, they 
also are really nothing but relations of magnitude and 
position between spatial constructions, relations, it is true, 
between different elements, but otherwise of the same 
nature as those assumed between the co-ordinates. Starting 
with a proportion between two marks x and jy, we do 
not arrive at really new marks, qualitatively incommensur- 
able with the first ; we advance merely from given homo- 
geneous relations to new homogeneous relations, and the 
derivability of the latter from the former, as well as their 
apparent novelty, depends merely upon the nature of space 
and upon the rules which geometrical perception has fol- 
lowed in reducing spatial relations to the universal laws of 
arithmetical quantities. These inferences therefore are far 
from meeting our requirement. The case is very different 
when we have to deal, not with mere spatial magnitudes, 
but with concrete objects,, in which a number of qualitatively 
incommensurable marks are united, and in which moreover 
science is unable to explain these primarily incommensur- 
able elements as merely different combinations of commen- 


surable ones ; in the face of these difficulties, thought will 
still have to look for a form which promises, approximately 
at any rate, the same advantages as those which mathematics 
with its easier problem offers in full. 

119. The group of mathematical forms of inference ends 
naturally here, with the emphatic recognition of the fact 
that the point which does not admit of being dealt with 
mathematically, the disparateness of marks, is precisely the 
point which we cannot avoid considering. The place of 
the equation will be taken externally by the form of defini- 
tion, for this combines a number of heterogeneous marks 
into a whole, but distinguishes in them a group of essential 
from another of unessential ones ; the former are regarded as 
containing the law for the combination of the whole, the 
latter as dependent on and determined by the former in 
accordance with that law. Lastly, this privileged group of 
essential marks can only be found by a comparison of the 
given concept with those which resemble it, and thus we 
are driven to systematic forms of grouping different things, 
and, primarily, to classification. 

C. The Systematic Forms. Classification. Explanatory 
Theory. The Dialectic ideal of Thought. 

120. When we began the account * of the formation of 
our concepts, we were already at the opening of the road 
which we have now to travel. We already recognised the 
matter of an idea to be a totality of different marks, united 
according to some definite rule that governed their con- 
nexion ; we already expected to find such a rule only in a 
group of marks possessed in common by different but com- 
parable ideas ; and already we noticed by anticipation the 
ascending scale of higher and higher concepts which results 
if we continue this process of comparing that which admits 
of comparison. I say, * by anticipation,' because the sug- 
gestion then made has not so far been turned to account 

1 [Sections 20-33.] 


in the later developments of logical activity. Judgments 
and syllogisms based on subsumption have only required 
us to consider the one relation which obtains between a 
concept $" and its proximate higher universal M- y there was 
no occasion for following up the relations of M itself to the 
higher grades of the series of concepts above it. For our 
only object was to make sure that a predicate P, which, for 
whatever reason, belonged to an M } must also belong to 
every S that falls within that M, and for this purpose the 
logical structure of M itself was to a great extent a matter 
of indifference. As middle term it bore the name of concept, 
but the character of a concept was in no respect essential 
to it; any simple mark, any sum of marks, whether com- 
bined under a definite rule, or merely brought together 
anyhow in thought, was good enough to constitute such a 
middle concept. It was only our concluding reflexions, 
which I shall not recapitulate here, that drew our attention 
to the necessity that the middle term should be a concept as 
we understood it at first, if we are to derive from it the 
right and obligation of a subject to possess the marks that 
it displays ; for it is only when thus understood that the 
concept really forms the complete rule under which the 
whole content presented by the subject coheres and is 

121. In saying this we are not simply returning to an 
earlier standpoint. In considering the most primary and 
simplest forms of thought, the logician can as a rule only 
elucidate their results by the use of examples which contain 
more logical work than he means them to illustrate. For 
these examples must be drawn from language ; and language 
is not the expression of a thought which has stood still 
where it began, but of the developed thought which has 
advanced by a multitude of successive steps beyond the 
imperfect results of its earliest endeavours, and which now 
conceals the recollection of them under the more elaborate 
setting which it has now given to its objects. And it may 


therefore seem as if our present problem, the formation of 
an essential concept, had been solved already in the above- 
mentioned passage; but it needed more than the logical 
acts which were then under discussion to generate the ideas 
which were there employed as instances ; such ideas could 
only arise by help of the processes which have now, familiar 
as they are, to be considered in their place in our system. 
Thought, in that earlier stage, met the countless multiplicity 
of composite images presented by perception, on the one 
hand with the desire to grasp each individual image as a 
whole whose parts are connected under a definite law, on 
the other with the consciousness that such a law could only 
be discovered by the comparison of many comparable indi- 
viduals and the retention of the common element in all. 
But such a comparison depended for the value of its results 
on one condition, namely, that the attention which executed 
it should be directed to a number of objects S, R, T, whose 
common element really consisted in the pervading law of 
their whole structure, and not to a number of others 7, F, 
Wj differing in all respects except the possession in common 
of a limited group of marks. Then, in the beginnings of 
thought, there was no logical rule for this selective guidance 
of the attention ; on the other hand, it was even then most 
effectively secured by the psychical mechanism, which makes 
those compound ideas reproduce one another predominantly 
in memory which are similar in the whole form of their 
connexion, and specially commends them to the attention, 
to the exclusion of those whose structure is dissimilar and 
whose agreement is confined to isolated groups of marks. 

122. In the actual course of its development, therefore, 
thought is first directed to those universal concepts which 
really contain the law for the complete formation of the 
individuals for which they are required ; it is not until it has 
some special motive in investigation that it frames universals 
in which things otherwise unlike are grouped under a fraction 
of similar elements. Thus when we were speaking of the 


first formation of concepts, the current instances of subordi- 
nation, e.g. of Caius and Titus to the concept of man, or of 
the oak and beech to that of plant, seemed to us quite 
natural and intelligible ; it was as if the mere direction to 
grasp the common element in the individuals was enough to 
put us upon the track of these really authoritative concepts 
M. And yet the same direction might equally well have 
led us to invent for negroes, coal, and black chalk a common 
name N y expressing the union of blackness, extension, 
divisibility, weight, and resistance : only the tendencies of 
the psychical mechanism favoured the first and hindered 
the second of these applications of the logical rule. 

123. These tendencies, which have hitherto unconsciously 
put us on the right way, we have now to translate into 
logical activity ; in other words, we have to become con- 
scious of the reasons which justify us in setting up a certain 
universal M exclusively as the authoritative rule for the 
formation of a number of individuals, instead of some other 
JVto which we might have been led by comparing the same 
individuals upon a different principle. Logic has shown us 
that a single form of interdependence between several re- 
lated points gives rise to different results ; we saw that the 
truth of the particular followed from that of the universal, 
but not that of the universal from that of the particular; 
and that while we could always infer from a definite reason 
to a definite consequence, a given consequence need not 
always lead back to only one reason, but might lead to 
several equivalent ones. Applying this to the organisation 
of a concept, we find in it certain marks a b c the presence 
of which has a determining influence upon the presence, 
absence, or modification of others, while the presence of 
these others, a /3 y, does not necessarily affect the former, 
but is equally compatible with different ones, / q r. This 
is the ground for the difference already mentioned between 
essential marks, a b , and unessential^ a J3 y ; it is only in the 
union of the former that we could expect to find the 


authoritative concept for the individuals compared, for it is 
only this union which determines the other marks and there- 
fore includes none but those individuals which are of 
kindred structure throughout ; the latter group of marks, on 
the contrary, would leave the former undetermined, and 
would therefore, if conceived as a universal, comprise a 
number of individuals otherwise entirely different. 

124. Our problem accordingly would be, to distinguish 
the essential marks from the unessential. This is easy so long 
as we have to do with objects which we can observe in 
different circumstances ; in that case the variable properties, 
which come and go as the conditions change, contrast of 
themselves with the permanence of what is essential. It is 
different when there is no possibility of such observation, 
and where, in the absence of varying circumstances, our 
object is to separate the essential from the unessential in 
permanent and invariable marks of the same concept : we 
have then to substitute comparison of different instances for 
observation of changes. Suppose a b c d to be the group of 
marks in one case of a given concept ; then, if in a second 
case of it d is wanting or is replaced by a quite different 6, 
it follows, on the assumption that all the parts of the con- 
cept cohere, that the remaining marks also experience a 
change; I denote the second case by a l b l c l b, to indicate 
that the Alteration of d to d does not cause the entire dis- 
appearance of any one of the marks in their universal sense, 
but only the transition of each from one of its possible 
modifications into another, the form of their combination 
remaining the same. In this case d does not belong to the 
essential marks ; it is the group A B C, including as modifi- 
cations a b c and a 1 b 1 c 1 , which regulates the organisation of 
the concept. But this first step informs us only that the 
marks united in A B C do as a fact remain together ; it 
does not show what internal coherence they have; the 
value of the several elements of the group may be very 
different ; it is possible that only A B or A C or B C 



contain the real law for the formation of the whole, while 
the third mark is merely a necessary sequel or allowable 
addition to the other two. As the mind is not yet in a 
position to investigate the actual object with all the ap- 
pliances of science, its only method of deciding this doubtful 
question is to continue the same process. We must com- 
pare ABC also with instances of the form A B T\ if 
the difference of the last mark is here too accompanied 
by no more than the previous deviation in the others, 
and the connexion of the whole remains the same, the 
coexistence and relation of A and B will be the dominant 
rule for the original a b c d^ or will represent that union of 
essential marks which makes the presence of the rest 
possible or necessary, or at any rate determines their 
amount, connexion, and relation to the whole. If we 
conceive this process continued, we find ourselves on the 
way to classification. We can now no longer confine our 
consideration to the individual if we would determine its 
concept ; that can only be done in this first of the systematic 
forms, that is, by investigating its nature in its relation 
to others, and judging from its position in an ordered 
series what degree of formative influence its several marks 
exercise upon its whole nature and behaviour. The au- 
thoritative principle of its formation will appear to us to 
lie in that inner circle of marks which, when \ve ascend 
through the next universal to higher and higher degrees 
of universality, remains together the longest and unchanged 
in its general form; and the only way to conceive com- 
pletely the nature of the particular is to think of this 
supreme formative principle as being specialised gradually, 
in the reverse order to the grades of universality, by new 
accretions which come within the influence of its reaction. 

125. The desire to get an explanation of the inner 
structure of the composite object by this systematic ar- 
rangement, lies at the root of all scientific classification, 
but is not equally satisfied by every form of it : before 


going on to consider the only form which will serve our 
purposes here, I will therefore briefly mention, as a pre- 
liminary, the artificial or combinatory classifications, which 
are designed specially to meet the general demand for 
clearness and summarisation, or certain particular require- 
ments of applied thought. We first by partition break 
up the content of a given universal concept M into its 
universal marks ABC...) and each of these by disjunction 
into its various modifications which cannot coexist in the 
same subject, A into a> a 1 a* . . ., B into b 1 & b* . . ., C intp 
& c* c 5 . Then, on the principle of the disjunctive judgment, 
every species of J^must possess one modification of each of 
the universal marks of M to the exclusion of the rest. 
If for the sake of simplicity we confine ourselves to two 
marks, of which the one, A, falls by disjunction into only 
two members, a and b, the other, B, into three, a, /3 and y, 
the binary combinations arrived at in the ordinary way, 
a a, a /3, a y, b a, b ft, b y, will comprise all conceivable 
species of M. Lastly, it makes the collective survey of 
them more easy if we place the modifications of the par- 
ticular mark which forms the basis of classification before 
the other marks, as was done above, or in the form 
M= a (a + ft + y) + b (a -f /3 -f y). The simplest instance of 
this classification is the arrangement of dictionaries; the 
fixed order of the letters in- the alphabet here gives the 
basis of division, not only in the first instance, but also 
for the numerous subordinate combinations contained under 
the head of each letter. The obvious advantage of this 
lexicographical classification is, that it gives a survey of 
the material, not only embracing all the words of the 
language, that is, all members of the object to be divided, 
but also making them easy to find, and this first advantage 
it shares with all successful attempts at artificial classi- 
fication ; but when we go beyond this we find that the 
degrees in which they contribute to the real knowledge, 
of their objects are very various* 

M 2 


120. We observe firstly that this method of combination 
only takes account of the marks of the given concept 
in their isolation, not in that mutual interdependence in 
which alone they really constitute the concept. Thus it 
is true that the sum of the combinations discovered in- 
cludes all species of M, but it may also include others 
besides them, which would be true species if the concept 
were merely the sum of its marks, but are not true because 
it implies their union in a certain definite form which 
these other species contradict. The concept of a triangle 
does not consist in the fact that we think three angles 
and three sides, but in the fact that three sides intersect 
one another so as completely to bound a plane space and 
by this very fact produce the angles. It is this connexion 
of the sides and angles which makes equiangular unequi- 
lateral and rectangular equilateral triangles impossible : in 
a classification by mere combination these would have 
found a place along with the equiangular equilateral, the 
rectangular isosceles, and other possible kinds. If the 
content of J/, as in this instance, is completely known 
and can be exactly constructed, these impossible forms 
are excluded by our knowledge of the fact, and the only 
use of including them in a provisional classification would 
be to stimulate attention to the nature of M, and to the 
reasons which make the valid kinds possible <and the 
invalid impossible. If on the other hand M is a generic 
concept derived from experience, the inner organisation 
of which can only be represented imperfectly by description, 
not exactly by construction, the species which we have 
not actually observed but should have been led to infer 
by the method of combination, remain doubtful; further 
observation may discover them, further knowledge of facts 
may show them to be impossible ; the use of assuming 
them provisionally may here also be to stimulate advance 
in orue of these two directions. 

127, If the method of combination, when applied to 


objects of experience, is liable to the uncertainty whether 
its results do not include more than the facts, it is true on 
the other side that, as ordinarily practised, it gives no 
guarantee that they exhaust the facts. It is beyond the 
power of human imagination to anticipate completely all 
the modifications to which a mark may be subject ; our 
attention will always be confined to those, / W 8 > which we 
happen to have observed ; another modification, / m , which 
does not come within the circle of our experience, will be 
missing in our classification along with all the species in 
which it may possibly occur, and this gap will not be filled 
up until our experience has grown. This is the ground for 
a logical rule, which is valuable when the decision of a 
question involves exhaustive knowledge of all the possible 
cases of some object Z ; the rule is to go on dividing and 
classifying them by simple contradictory opposition. The 
sum of all possible cases of Z is always of the nature Q or 
of the opposite non-<2 ; the cases of the form Q are always 
either R or non-^?, those of non-(? always either S or 
non-; so that at whatever point the division is broken off, 
all possible cases are included by it. Such a method, 
indeed, is only fruitful when we are so happy in our 
selection of the first opposites Q or non-Q, or of all the 
subordinate opposites in the same grade, S, non-S, R, etc., 
that we can show without much trouble whether or no the 
characteristic in question Z is exhibited in each of the 
alternative cases. 

128. It is moreover evident that in classification by com- 
bination there can be no logical rule obliging us to employ 
certain marks at the top as bases of division in the principal 
groups, and certain others lower down in their subdivisions. 
So long as the concept M which is to be divided is con- 
sidered merely as a sum of its marks, without regard to 
their mutual relations, any one of them has a right to form 
the principal division by its modifications, and any other 
may be subordinated to it as basis of a subdivision. The 


obvious disadvantages of this uncertainty are avoided in 
practice by concomitant reflexion and an estimate of the 
different values of the marks, based upon a knowledge of 
the facts or a right feeling, often merely upon an instinctive 
taste : all that logic can contribute to these precautions 
is the general direction not to choose as bases of division 
notiones communes, i.e. marks which are known to occur in 
the most different objects without exercising any recognisable 
influence upon the rest of their nature. The positive 
direction answering to this prohibition, viz. how to find the 
decisive bases of division, logic leaves entirely to be given 
by special knowledge of the matter in question. And as 
regards complex concrete objects at any rate, so long as 
fundamental divisions were based upon single marks, the 
specialist has always been open to the criticism that he 
sometimes removes closely related species to different and 
often very distant parts of the system, while he brings 
others which are totally and strikingly unlike into surprising 
proximity. This is quite intelligible when we consider the 
different influence which the marks have on the structure 
of the whole concept. There is no reason, for instance, 
why the mark B, so long as it occurs in the modified 
form $, should not conspicuously affect the formation of the 
whole, and in that case all the species under the head of b 
will remain connected in form; but the same mark may 
entirely lose this influence as soon as it enters into the 
group of marks in the modified form /3 ; then the species 
under the head of /3 follow all the variations due to the now 
influential difference of the other elements A C >, and 
examples of M^ otherwise most unlike, now find themselves 
in the closest proximity. This is what happened to the 
Linnsean system, which selected the number of stamens as 
the basis of division ; the result of this view was, that in the 
cases where the whole organisation of the plant made the 
stamens of importance, the related species were brought 
together; where this was not the case, they were separated, 


and different species were united. An instructed taste will 
partially obviate this evil also, by selecting different bases of 
division for different sections of the whole system. Nothing 
but an unseasonable logical pedantry could require that a 
system which had begun by dividing its whole object-matter 
according to the modifications a b c of one mark A, should 
go on to arrange all the groups formed by #, ^, or <r, accord- 
ing to modifications of one and the same second mark B ; 
it may be that the variations of a mark C are exclusively 
of importance for the group with #, and those of a fourth 
mark D for the group with b^ and the classification which 
proceeds upon this view approaches by that means, and by 
that means only, to the real essence of the thing. The risk 
which such a method runs of not discovering all the species 
completely, must be avoided in some other way ; classifica- 
tion does not create the complete material, but assumes its 
completeness to be guaranteed elsewhere. 

129. Classifications would belong entirely to applied 
logic if they aimed at nothing more than complete sum- 
marisation, such as is required either when we wish to deal 
with a subject practically or when we are just beginning to 
consider it logically. But they do more than thus merely 
prepare the ground; they themselves represent a logical 
ideal, which has its necessary place in the systematic series 
of the forms of thought; the very fact that a manifold 
material has been brought into the connexion of a classified 
system, is of itself supposed to tell us something as to the 
nature of each and all of its members, and not to be a 
mere preliminary to future enquiry. This appears in the 
objections which we make to forced classifications ; we not 
only require the lines along which we must look, in order 
to find a particular species, to be precisely laid down 
beforehand in a series of concepts, but we expect the 
actual places in which the several species are found to 
correspond in position to the affinities of the species them- 
selves. For practical purposes any order will serve that is 


handy for the person who is going to use it, but the order 
which logic demands must be true to the facts. Now if we 
wish to form a complete idea of any composite object, it 
does not matter with which of its parts we begin, provided 
only that the order in which we add each new part is 
adapted to the particular point with which we have chosen 
to start : any idea of a given content so arranged forms a 
concept of it, sufficient to distinguish it from others and to 
show what it is itself. Amongst these various concepts of 
the same M there is one distinguished from the rest by 
having for its starting-point the law which determines the 
order of all the other marks, and this is the one which we 
try to find. We have already given the name of c constitu- 
tive' to such a privileged concept ; it might also be called, 
in opposition to the mere conceptual form in general, the 
logical Idea^ of the object, or, in the vernacular, its thought \ 
for it is thus that our language distinguishes the 'idea' of 
plant or organism, as its formative law, from the concept of 
it, which merely comprises the sum of the necessary marks 
and the form in which they happen to be combined. 

130. It will help us to realise what has just been said if 
we mention here two incidental notions which always attach 
themselves readily to this search for the Idea of an object, 
conspicuously in the attempt of naturalists to improve the 
artificial classifications of plants and animals by Deference 
to their natural affinities. In these cases we are prone to 
regard the universal Idea of animal or plant as a living and 
operative force, whose unvarying and consistent activity 
gives rise to a series of different forms, accordingly as 
external conditions determine one or more of its points of 
incidence and oblige it to change correspondingly the 
whole course of its action. Another way in which we are 
equally prone to regard it is as an unvarying end, which 
regulates its modes of operation according to the relations 
in which it finds itself placed, and in the different forms 
1 [ Idee.*] 

Chap. III.] THE 'IDEA' AS END. ,169 

which it is thereby compelled to assume realises one and 
the same purpose in various ways or with various degrees of 
completeness. From this point of view the different species 
classified together express the result of the interaction 
between the universal idea and the particular relations, with 
which as universal it has nothing to do. It will be admitted 
that these ways of looking at the matter place it before us 
in a clear and vivid light, but it will also be objected that 
they are both quite foreign to logic. The objection is 
unanswerable; our intention however is not to turn the 
ideas of active tendency and purpose to account for the 
benefit of logic, but to show that even in their proper place 
they only have meaning on the assumption of a purely 
logical notion, which we will now explain. If it is to be 
possible for the same end to be fulfilled under changing 
circumstances, it must also be possible to express its con- 
tent by a group of ideas, Z, in which these different forms 
of fulfilment cohere as possible species, and from which 
they necessarily result if each one of the marks of Z and 
each of their mutual relations is successively subjected to 
all the changes of which, as parts of Z, they are respectively 
capable. If, again, an active tendency is to change its 
activity under varying conditions and to manifest itself in 
new results, the combination of forces in which it consists 
must hp expressible by equations, from which all these 
new formations necessarily follow as soon as we give the 
quantities entering into the equations all the values suc- 
cessively which their natures allow. Activity, then, whether 
intentional or unintentional, never produces anything but 
what is abstractedly possible to thought, and this becomes 
necessary to thought as soon as we affirm one of a number 
of related points upon which the rest depend. It is this 
which we have in view here : we regard the idea for which 
we are looking, neither as the intention of a reflective 
consciousness striving for fulfilment, nor as an active force 
which causes its results, but merely as the conceived or 


conceivable reason, the consequences of which under certain 
conditions are the same in thought as those which must 
follow in reality, under the like conditions, from an intelli- 
gent purpose or a causative force. Keeping this in mind, 
we may tolerate a phraseology which imports into logic the 
idea of an end or of a tendency to development : it will 
nevertheless be better to avoid these expressions, and not 
to use what is found only in the real world as a name 
for the mere reason upon which in thought the reality 

131. Another point which logic cannot neglect may be 
introduced here as a sequel to these accessory notions. We 
are not surprised in a self-realising tendency if, under certain 
conditions, it fails in its endeavour ; and we find it intelli- 
gible that an end should be attained under different circum- 
stances with different degrees of completeness. Thus both 
these notions very naturally give rise to the assumption that 
different realisations or examples of the formative idea are 
of different values, and that they are not merely co-ordinated 
in a general way as species under the universal concept of 
their idea, but form within this co-ordination an ascending 
or descending scale in which each one has its uninter- 
changeable place between certain others. The attempts at 
natural classification, which endeavour to satisfy our modern 
requirements, are dominated throughout by this thought ; 
and it remains to show that this familiar tendency to pass 
from classification by mere combination to classification in 
the form of a developing series, is justified on general 
logical grounds, and that this is the place to justify it. 

If, as is too often the case at the beginning of logic, we 
regard a concept M merely as a sum of marks universally 
expressed, there is no sense in rating one of its species 
higher than another. Every .S" either contains all the marks 
of its universal M y and in that case it is a species of it, or it 
does not contain one or other of them, and then it is, not 
an imperfect species, but no species at all of M. But living 


thought in actual practice is far from acquiescing in this 
hard antithesis; it distinguishes species which correspond 
or are adequate to their generic concept in various degrees. 
The possibility of making this distinction depends primarily 
upon quantitative measurements to which the several marks 
and their relations are possibly or necessarily accessible. 
The structure of generic concepts, incalculably as it varies 
in particular instances, agrees in the main in containing a 
number of parts or related points, each comprising a group 
of simple marks and standing to the others in all sorts of 
relations. By * simple marks' here I mean, not only 
sensible properties such as red, sweet, hot, but others also 
like heavy, extended, irritable, which, though no doubt they 
contain the result of previous observations of complex 
modes of behaviour, contain it in so simple a shape that 
our logical imagination has long accustomed itself to attach 
them to their subjects as stable and simple predicates. To 
all these elements of the concept quantitative differences 
extend. No mark of any one of its parts is conceivable 
without a definite degree of its specific kind of intensity, 
and the degrees may vary infinitely; the number of the 
parts themselves can, like every number, be increased or 
diminished, and every part moreover can alter its logical 
value by expanding the simplicity which belongs to it as 
a member of the genus into a complex organisation of its 
own inner nature; and lastly, every relation between the 
various constituents of the concept varies in value according 
to the value of those constituents, or admits of greater or 
less closeness according to some standard of its own. The 
joint effect of all these possibilities of variation is to produce 
a number of species noticeably different. If we suppose 
that when a mark P of the generic concept M assumes the 
value /, the influence which it always exercises upon the 
other marks is so intensified as entirely to change the form 
of the whole content of M^ the resulting species will no 
longer be a species of M, but of some other genus N. And 


those values of P which approach this decisive limit but do 
not reach it, will produce forms which still fall under the 
genus M y but approximate gradually to the structure which 
is characteristic of N. It is upon this that the difference is 
based between species which are more and less appropriate 
or adequate to their common generic concept ; each species 
is in a certain respect more perfect the farther it is from 
passing over into another genus, and that is the logically 
most perfect whose divergences from all proximate genera 
make up the greatest total amount. 

132. I believe I am justified in saying that this point of 
view belongs entirely to logic, and is independent of the 
views which we may form on other and material grounds 
as to the value, meaning, and function of anything which 
has the law of its existence in a generic concept. I will 
therefore illustrate it by examples which are not affected 
by these incidental considerations. The equation of the 
ellipse, a 7 ' y 1 + b* x z = cP 2 , leaves the two axes a and b to 
be chosen at pleasure, and the formula claims that it will 
always produce an ellipse whatever values we may assume 
for a and , and even therefore if one of them be assumed 
to = o. But in that case the curve passes into a straight line, 
and the result which this value gives falls accordingly under 
the concept N, that of straight line, which is different from 
that of the ellipse. But this example shows at tfie same 
time, what we did not choose to assert universally above, 
that the extreme species of a genus M^ when produced in 
this way, not only must belong to a new genus N^ but may 
also continue to come under the former genus M* It is 
true that the central equation of the ellipse can tell us 
nothing about this case when b = o, because it then ceases 
to indicate a curve. But there is another expression of the 
essential formation of an ellipse which is still valid ; namely 
the rule that the sum of the radii vectores, drawn from two 
fixed points on the major axis to one and the same point on 
the periphery, is constant and equal to the major axis. In 


the present case where the ellipse has shrunk into a straight 
line the two extremities of the line are identified with those 
two fixed points, the foci of the ellipse, and for every inter- 
mediate point c we have the sum of the distances a c -f c b, 
that is, the sum of the two radii vectores^ equal to the length 
a b of the straight line. 

If a heavy rod of the fixed length a b stands with one end 
a on a perfectly smooth horizontal surface, and with the 
other b leans against a perfectly smooth vertical wall, the 
pressure of its weight makes equilibrium impossible and it 
falls. An easy calculation shows that the path described 
during its fall by any point C in its length is an ellipse. At 
the same time it is clear that the end b must slide down the 
wall in a straight line perpendicularly, while the point a must 
move away upon the smooth surface in a straight line hori- 
zontally. As then every point in the line is affected by the 
same group of conditions, these rectilinear motions also 
must be regarded as specific forms of the elliptical path 
required generically by those conditions. They are in fact 
the two extreme cases which we get if we make first one 
and then the other axis = 6 ; the end of the rod then 
moves in a straight line in the other axis. The middle 
point of the rod supplies another singular case ; the axes of 
its elliptical path are equal, and thus it describes the arc of 
a circle* The nature of the problem before us compels us 
therefore to conceive the circle as a species of ellipse, and 
the central equation which we have mentioned makes it at 
once clear how this is possible. This example therefore 
shows us that by changes in the quantity of one of their 
parts the species of a genus M approach gradually to the 
formative law of another genus, and that there may be 
limiting instances which are species both of M and of JV, 
because they satisfy the requirements of both concepts ; by 
merely examining the actual constituents of such a limiting 
instance it is impossible to tell by which generic law its 
form is, strictly speaking, determined ; in the present state 


of our knowledge this question is decided upon incidental 
grounds of various kinds. 

133. On the other hand, these examples leave an am- 
biguity which must be removed in regard to the standard 
by which we measure the degree of perfection, or, to put it 
shortly, the height of each species. Mathematical figures 
have no history telling of their life and growth ; being 
merely legitimate possibilities of thought without real exist- 
ence, they can be produced for our imagination in the most 
various ways, and it is in the abstract indifferent, and in any 
particular case depends on the nature of the problem in 
question, from what point we begin their construction, or 
under what generic concept, what universal rule of con- 
struction, we bring them. If we look at them, not geome- 
trically, but aesthetically, I mean if we attend to the total 
impression of the figure as it is, not to the way in which it 
came into being, circles and straight lines contrast decidedly 
with ellipses. In the impression of the ellipse as we per- 
ceive it the inequality of axes is a necessary element ; on 
the other hand it is true that the greater this inequality is, 
the more does the curve approach the extreme forms which 
we wish to exclude, that of the two straight lines which 
coincide with one or the other axis. The characteristic 
impression of the genus would be best produced by an 
ellipse equally removed from the equation a <=cythat of 
the circle, and from the equation <z <=#, that of the straight 
line. By combining both equations we might define the 
condition of this impression by saying that one axis must 
be double the other, and this would be tolerably correct ; 
only that a thing cannot be mathematically determined 
which does not depend simply on mathematical laws. Our 
logical imagination is dominated in every direction by simi- 
lar tendencies. Nothing is commoner than for a person 
who speaks of a quadrangle to mean really a parallelogram, 
or often even a square ; and this inexactness in expression 
is very natural ; the imagination wants to realise the concept 


in perception, but can only hold one image at a time, and 
it therefore chooses the image which is logically most per- 
fect ; and it is the fact that the parallelogram, by increasing 
inequality either of the sides or of the angles, continually 
approximates to the ultimate form of the straight line, in 
which all the four sides coalesce. The observation of 
natural objects evinces the same tendency; we always re- 
gard as the typical and most expressive examples of each 
genus those species in which all the marks are at the 
highest value which the combination prescribed by the 
genus allows, in which therefore no mark is exclusively 
prominent and none is reduced to zero, but all combine, as 
far as possible equally, to produce the impression of stable 
equilibrium in the whole. 

134. I will here repeat an observation which I made 
before. I am not afraid that anyone will criticise this mode 
of estimating the relative height of species on the ground 
that it has nothing to do with logic; its defect is rather 
that it starts from inadequate logical grounds, and does not 
adapt itself sufficiently to the nature of its objects. To put 
it shortly ; that the highest perfection of a species depends 
upon the equilibrium of its marks as described above, is the 
opinion to which we must come on purely logical grounds, 
so long as we have no positive knowledge to supply us 
with some other standard of measurement based upon the 
essential characteristics of the genus in question. It may 
lie in the nature of things that a genus M can not maintain 
this equilibrium of marks, but is destined by diminishing 
one and intensifying another to pass over into another 
genus N\ in that case its species will be more perfect 
in proportion as they approach more nearly to this point of 
transition at which they cease to belong to their own genus. 
We find that the most important attempts at natural classifi- 
cation are deeply imbued with this idea of a destination to 
be attained, which is constantly impelling the several genera 
to advance beyond themselves ; I therefore introduce it here 


intentionally, in order to notice its significance for logic, 
with which in itself it has nothing to do. We have already 1 
separated the idea of productive activity from the concept 
of tendency, and the idea of purpose from the concept of 
end; we must in the same way separate here the idea of 
obligation from the concept of destination. Everyone will 
see that the effect of this separation is to do away with all 
that is characteristic in the meaning of these three concepts ; 
but this is just what we are aiming at. It is not the concept 
of destination itself which we are importing into logic, but 
merely that of the logical relation upon which it is essen- 
tially based, and of which it is itself so graphic an illustration 
that we can hardly avoid the term as a figurative expression 
of the logical truth. A destination, then, which has to 
be reached, differs from a final state which merely happens 
to be reached by some process of change ; in the former 
case the group of marks which characterises the end attained 
contains also the authoritative principle upon which the 
marks are connected and upon which they change as they 
do ; in the latter, the processes which lead to the end may 
take various directions, forwards and backwards, to this side 
and that. Bearing this in mind, we can no longer doubt as 
to the purely logical sense of the word when we speak of 
a 'destination' to which the several genera have to approach. 
Hitherto we have looked upon the generic concept M as 
the ultimate authoritative principle which regulates the 
series of its species, and that species therefore as the highest 
which exhibits this concept in the most perfect equilibrium 
of its marks; now we are reminded by a consideration 
originally foreign to logic, that the case may be different, 
and that the formation of the series of species in M need 
not really depend on anything in the generic type of M 
itself, such as could be discovered by merely examining its 
own constituent marks ; that, on the contrary, the formation 
of this genus is not rightly explained until we compare 
1 [Above, 130.] 


it with another genus N into which it passes, and with 
a third L from which it came by a similar transition, and 
these again with those which went before and came after 
them; not till this comparison has been made do we get 
the direction in which the progress towards perfection takes 
place within a higher genus Z, of which L M N are species ; 
then, in the series of species in any particular genus M^ 
those species will be the highest which have advanced the 
farthest in the direction in which ^/as a whole is develop- 
ing towards the most perfect expression of the higher Z 
which includes it It remains to show that this line of 
thought, to which we were originally led by an extraneous 
suggestion, has its necessary place here in the internal 
economy of logic. 

135. It is scarcely needful, however, to show this. We 
have seen that we could only produce the universal concept, 
which includes a number of individuals under it, by uniting 
their permanent and common marks ; then we saw that 
this constant group of marks might contain elements of 
very different values, and in order to separate those which 
are not only constant but contain the rule to which the rest 
must conform on joining them, we had to compare the 
universal already found with other universals, and species 
with species ; that which still cohered in this wider field of 
change we regarded as the true essence of a genus M y 
the species of which were to be ranked higher or lower 
in proportion as they realised it more or less perfectly. 
But this process has no natural ending ; the same questions 
continually recur ; the marks which constitute M will them- 
selves differ in value, and the only way to distinguish the 
essential from the unessential will be again to compare M 
with L and 7VJ to form the higher genus Z from the law 
which persistently governs the formation of them all, and to 
measure the value of ML IV, as well as that of their several 
species, by the degree in which they realise this law Z, 
instead of by the degree in which each species expresses the 

LOGIC, You I. N 


more special law of its own proximate genus. This progress 
might go on to infinity, or to the point at which we suc- 
ceeded in finding a highest ideal A, exhibiting the mode of 
connexion to which all kinds of existence, real and think- 
able, must conform : from this A a classification might be 
derived in the form of a development which evolved from 
itself the whole content of the universe, and this develop- 
ment, if it were possible, would give the only logical security 
that every species had a place in the series of cognate 
species answering to the degree of essence which it ex- 
pressed. Thus the problem of natural classification leads 
of itself beyond the isolated treatment of a particular problem 
to the systematic organisation of the whole world of thought. 
And this tendency has in fact guided the most important 
attempts at such a classification. Those who have wished 
to exhibit the development of plants or animals in an 
ascending scale, or the events of history (for this form 
of thought claims to apply to processes also), have always 
been obliged to justify their selection of a particular standard 
for measuring the increase in value of the several members 
of the series ; this justification they have always had ulti- 
mately to find in certain general views as to the meaning of 
all being and process, views which are either formally ex- 
pressed at the very beginning of the enquiry, or make 
themselves tacitly felt throughout it as a guiding principle. 

136. Natural classification, then (to sum up under the 
traditional name the procedure just described), differs from 
combinatory or artificial classification in taking account 
of the mutual determination of marks which in the latter 
received only subordinate attention, while in its result it is 
distinguished by its serial form, in which the members are 
not merely placed side by side, but follow each other in a 
definite order leading from the province comprehended or 
dominated by one species into that of another : this order 
begins with those members which answer least to the logical 
destination of the whole system, and ends with those which 

Chap. III.] TYPE AND IDEAL. '179 

express in the most complete and pregnant way the fulfil- 
ment of that destination. But the simplest case here 
supposed, that in which the series has only one direction, 
is not necessarily the only one. In the first place it is 
conceivable that single marks in each species may vary 
without altering the characteristic structure of the species 
at all, so far at least as we can see : in that case the 
different instances of this species are equal in value, and 
the series may thus be increased in breadth by co-ordinated 
members without growing in length. It is also possible 
that, owing to different or opposite variations in several 
marks, a species M may not only pass over into one 
proximate species N, but branch out into several, N, O, Q, 
with which it has equal affinity and which contribute 
equally to carry out the general development ; these will 
then become starting-points for new series, which either 
continue side by side or subsequently coalesce again 
somehow with the central series. Thus the form of natural 
classification in general is that of a web or system of series ; 
even the culminating point of the system need not be a 
strict unity, for the most perfect attainment of the logical 
destination is compatible with a variety of precisely equiva- 
lent forms. 

137. As the occasion suggests it, I will mention two more 
concepts in frequent use, which may find a logical explana- 
tion here. The new kind of value which each species 
acquires in proportion as it approaches the end to which 
they are all developing, does not exclude the other kind 
which we mentioned earlier, depending on the equilibrium 
which it exhibits in the marks of its proximate genus. The 
two values subsist side by side, though the one impairs the 
other. We feel the conflict between them in our aesthetic 
judgment of phenomena. Every species which expresses 
its genus in the stable equilibrium of its marks, impresses 
us as perfect, relatively or absolutely : such a species forms 
the type of the genus, that type which is the indispensable 

N 2 


though not the sole condition of beauty in the beautiful, 
and which gives even to what is abstractedly ugly the 
formal right to a subsidiary place of its own in artistic 
representation. On the other hand, species in which this 
equilibrium is disturbed by approximation to an end higher 
than can be attained within the limits of the genus, give us 
the ambiguous impression which we call ' interesting,' like 
dissonances in music, which do not satisfy us but prepare 
us for a higher satisfaction. Ideal as opposed to type would 
mean a phenomenon in which the equilibrium of marks 
required to make it typical coincides happily with the 
highest development in regard to its logical destination ; 
logic does not exclude the possibility of such a coincidence, 
and art may perhaps find it realised or be able to realise it 
in a phenomenon in repose, though more probably only in 
some situation of the phenomenon. 

138. Lastly, it will be asked, how classification by de- 
velopment reaches its required conclusion, the certainty, 
namely, that it has really found that supreme law or logical 
destination which governs the particular object or the 
universe at large. To this we can only answer, that by 
way of mere logic it is quite impossible to arrive at such a 
certainty. The form of classification by development, like 
all logical forms, is itself an ideal, an ideal which is 
demanded by thought, but which can only be realised, so 
far as it can be realised at all, by the growth of knowledge. 
Nor indeed is this an exceptional condition, such as would 
lay this first of our systematic forms under a disadvantage. 
The judgment also enjoins a connexion of subject and 
predicate which thought has to make if it wishes to come 
into contact with its object in its own way; the hypothetical 
judgment, for instance, tells us, that only by annexing a 
condition to the subject S is it possible to ascribe to it a 
predicate P which is not already contained in the concept 
of -S"; but logic does not tell us what condition x is necessary 
in order to secure this particular P for this particular Sj it 


waits for special knowledge to put its injunctions into 
practice. The theory of the syllogism also teaches us how 
to draw conclusions when the premisses are given, but it 
does not give us the premisses, nor does it guarantee their 
truth, except so far as they may themselves be conclusions 
deducible from other premisses ; these latter then serve as 
the material given to thought, and lead back finally to some 
truth which is no longer logically deducible. Similarly 
all that the theory of natural classification asserts is, that 
every group of complex and coherent objects, and therefore 
(since everything coheres) the whole realm of the real and 
the thinkable, must be regarded as a system of series in 
which concept follows concept in a determinate direction ; 
but the discovery of the direction itself, and of the supreme 
directing principle, it leaves to positive knowledge to make 
as best it can. 

139. It is not this objection, but a difficulty of another 
kind, which obliges us to continue our enquiry. The 
difficulty will be most easily understood by reflecting on 
the place which classification occupies in our system. As 
a certain arrangement of concepts, it answers primarily to 
our first main section, the theory of the concept itself; 
but we were obliged to pass on from the concept to the 
judgment, for we found changes in the content of thought 
which could not be apprehended by conception alone ; on 
the contrary, the concept presupposed relations between 
its marks which it needed the judgment to interpret clearly. 
Classification answers moreover to the first form of judg- 
ments, the categorical ; as in these the subject simply had, 
assumed, or lost its predicates, so here the supreme authori- 
tative concept appears by itself as the sole producer of all 
its species, as the source from which they emanate. But 
the hypothetical judgment met the categorical with the 
objection that a single subject S cannot by itself give rise 
to any multiplicity ; and, similarly, all theories of emanation 
will have to ask themselves the question, what second 


condition it is which makes their first principle develope 
at all, and whence come the data in reaction against which 
it is obliged to expand into these particular forms and no 
others. A corresponding advance is called for here ; and 
it will prepare the way if we consider it in still closer con- 
nexion with the characteristics of classification described 
above. We made it an objection to artificial classification 
that it may lead to impossible instances, while in classi- 
fication by development we gave proportionately more 
attention to the mutual determination of marks ; we as- 
sumed that a change in one mark reacts upon the rest, 
that through this change one concept passes into another, 
and that one species answers better than another to its 
concept. This clearly implies that in the formation of its 
species the concept depends, not only on itself, or, in 
figurative language, on its own purpose, but also on 
another power which determines what kinds of realisation 
of that purpose are possible or impossible, adequate or 
inadequate. This power we have to investigate. 

140. The problems of thought are not completely solved 
until it has developed forms for the apprehension of every- 
thing which perception offers to it as an object and stimulus 
of its activity. This requirement, that all thinkable matter 
should be included, is not satisfied by classifications. Their 
natural objects are always those stationary generic forms 
with stereotyped marks, which we believe ourselves to have 
before us in perception as fixed points for manifold relations, 
but which are far from constituting the whole of what we 
really perceive. The several genera are not found in reality 
arranged in the system in which classification exhibits them ; 
as they actually appear they are always realised in number- 
less individual instances, separated in time and space, and 
subject to continual change both in their own conditions 
and in their relations to one another. Even if we admit- 
that the nature of each generic concept contains the law 
which every instance of it will obey // it occurs under 


certain circumstances, yet there is no reason in the concept 
itself for the hypothetical addition which we make, neither, 
that is, for the presence of that instance at the time and 
place at which it is present, nor for the occurrence or 
non-occurrence of those particular circumstances. Thought, 
therefore, does not embrace in the form of classification 
all that there is for it to embrace ; and that which appears 
here merely as an incidental stimulus to the universal 
concept to produce this or that species of itself, must 
also be taken account of as an essential part in the or- 
ganisation of the thinkable world as a whole. 

141. These considerations are not disproved by the fact 
that, as we observed before, classification by development 
may extend, not only to generic forms of the real and 
the thinkable at rest, but also to progressive processes. 
For when it is attempted to represent history as a deve- 
lopment, the question what it is which makes process 
process, the coming of one state into being out of another, 
equally escapes the grasp of logic. When they are re- 
flecting on the past or forecasting the future, these specu- 
lators may picture to themselves certain situations as 
temporary states of equilibrium, which they assume to 
follow one another on the stream of events in a fixed and 
necessary order; but how the transition from one to 
another . actually comes about, they cannot tell us. Nor 
could they do so even if they undertook the endless task 
of dividing the interval between two such states of equi- 
librium into an infinite number of stages ; they would be 
able to show that the concept of each stage, when it is 
reached, is preliminary to the concept of the next, but 
they could not show how the reality which this concept 
expresses brings the reality expressed by the other in its 
train. We must reflect moreover that in the real world 
pure concepts do not occur or develop themselves, but 
only particular examples of them, each with all its marks 
specifically modified in a way which its concept allows 


but does not necessitate. Not only therefore does the 
process of becoming remain a mystery which classification 
cannot explain, but the result of the process results, not 
from the concept of the stage preceding it, but from that 
particular realisation of the concept of which also classi- 
fication takes no account. All the attempts both of ancient 
and modern times to derive the world by way of emanation 
from an original concept, are subject to the same defect. 
If their original concept is really nothing but the pure 
thought of a relation which certain elements not yet named 
necessarily imply, all that they can derive from it will 
be certain forms, likewise universal, in the shape of possi- 
bilities, or, as I have no objection to say, necessary re- 
quirements, which in the event of being realised must be 
realised in a certain way ; but they have no means of 
deciding what this way will be, or of showing where the 
desired realisation will come from. If on the other hand 
their original thought expresses a relation between elements 
not unnamed but definitely characterised, and is endowed 
itself with the impulse to development which those elements 
do not supply, in the shape of an inherent restlessness 
which drives it to evolve its consequences, this is only 
to admit that the complete form of each new stage of 
development does not depend only on the concept of 
the preceding stage, but on the special form in wh,jch, as a 
fact, but without any reason, that concept had already 
realised itself. It is to admit, in other words, that along- 
side of their categorical development by emanation of the 
concept out of itself, another power is also at work; this 
power, which their theory entirely disregards, consists of 
a sum of authoritative hypothetical relations, which ordain 
that if the marks in a given concept have as a fact a 
certain value, and if certain conditions act upon these 
marks, the form of the new resulting concept, the new 
stage of emanation, is then, but also not till then, com- 
pletely determined Lastly, if we compare the theory of 


emanation with the method of the inferences by sub- 
sumption, we may say shortly that what it lacks is the 
second premiss, by which alone they produce from the 
universal major the comparatively more special conclusion. 
These subsidiary ideas, which are here only tacitly pre- 
supposed, logic has to supply explicitly : it cannot stop at 
a classification based upon concepts, but must point out 
also the legitimate connexion of the judgments which ex- 
press the power of a mark already in existence to determine 
another which is to come into existence out of it. 

142. But it is not necessary to confine ourselves to that 
side of classification where it fails to give a complete solu- 
tion of the problem of thought ; the attainment of its own 
more limited end implies the same tacit assumptions. Each 
of the generic concepts classified is necessarily composed 
of marks which occur in other concepts as well. It would 
be lost labour to construct a scale of genera L M IV, if L 
had marks which were heard of nowhere else in the world, 
and M and N were distinguished by similar uniqueness. 
The marks must rather be looked upon as building-stones 
tying about ready for use ; they have to be cut differently 
according to their different positions, but they are all of 
commensurable material, and it is only the different ways of 
using it which give rise to concepts of different structure. 
Now in ^classification by development the marks united in 
the same generic concept M are spoken of as mutually 
determining each other ; a change in one is followed by 
changes in another ; and the progress of these changes not 
only produces the several species of the genus 3/ 5 but leads 
beyond them into the genus N. What rules can this 
influence of one mark on another follow but such as involve 
a universally valid relation between the natures of these 
marks"} And as the marks themselves hold good beyond 
the limits of the particular concept M, this relation also 
must be independent of M. The formation, therefore, of 
the several species of M, their possibility or impossibility, 


and ultimately the possibility or impossibility of M itself, all 
entirely depend on what is allowed or not allowed by these 
universal taws of connexion between the marks. Accord- 
ingly, the classification of concepts cannot fulfil even its 
own proper function without presupposing a system of 
judgments or universal laws regulating the admissibility, 
mode of connexion, and mutual determination of all marks 
which are to be united in this or that generic concept. 

143. I must mention here an apparent contradiction, the 
removal of which will conclude these preliminary considera- 
tions. We have already, in treating of the form of propor- 
tion, spoken of the necessity of this mutual interdependence 
of marks ; we there corrected ourselves by saying, that when 
a constant relation exists between two marks, the measure 
of their interaction is not found in the marks as such, but in 
the nature of the whole in which they occur or in the con- 
cept of that whole. We seem here to be retracting this 
statement, but we are in fact confirming it. For the very 
point which we have now made clear is, that the content of 
the concept, to which we there transferred the decisive in- 
fluence, is nothing but a number of marks, each extending 
beyond the concept itself, and all connected in it in a 
definite way. Between these marks, as we saw, different 
relations are possible ; it may happen that the idea of one 
involves that of another ; in that case every subjeqt which 
has the first will have the second also ; or it may be that 
two marks exclude each other as contrary and contradictory 
members of a common element, and in that case there is no 
conceivable subject in which they can exist together ; 
between these extreme cases lie others, in which, without 
any similar logical grounds, we perceive two marks to be 
combined as a fact, but the value of the one does not 
always imply a like value in the other. These are the cases 
to which our observation above applied; for the reason 
which narrows the range of this variation, and fixes the 
precise proportion in which two marks determine each other 


m any particular object, lies in the simultaneous presence 
of all the other marks, in the values and the mode of their 
combination. What was undecided in the relation of the 
two is decided by their relations to the rest ; if the different 
equations, by which we may suppose the latter relations to 
be expressed, are only satisfied by one value of each of the 
marks, the formation of the whole is completely defined ; 
where the number of equations is not enough for this, the 
whole is still partially indefinite, and exhibits a universal 
concept in which there is still a possibility of different 
species. Thus it is true that the concept determines for its 
subordinate species the proportion in which each pair of 
marks condition one another ; but it only does this in virtue 
of the ordered sum of its other marks, and so far as these are 
known to have definite values. Our method, in fact, has 
always been based upon this supposition. In proposing to 
classify a generic concept by developing its species out of 
it, we have always had to assume that certain of its universal 
marks are already defined by their places in the series ; not 
till then could the rest acquire that definite character which 
was necessary to complete the distinction of one species 
from another. In the concept itself the existence of this 
primary definiteness, of which the rest was a consequence, 
was only a possibility ; its realisation was assumed in 
thought .independently of the concept. 

144. If we sum up these considerations, we may say that 
every individual and every species of a genus is what it is 
through the co-operation of the complete sum of its con- 
ditions ; these conditions consist in the fact that a number 
of elements or marks, which might also exist in separation, 
are as a fact given in a certain combination, which might 
conceivably be different, and each with a certain quantita- 
tive value, which is one amongst other possible values. 
From this given union of conditions, according to universal 
laws which hold good beyond the limits of these elements, 
this perfectly definite result follows. Every such result, 


when it is once there, can be compared with others, and co- 
ordinated with them as species with species or subordinated 
to them as species to genus; but these concepts, which 
hitherto we are considering as the key to the understanding 
of the structure of their subordinates, must not be credited 
with any mysterious and authoritative power, beyond the 
fact that they are condensed expressions for a definite union 
of separable elements, which act and react upon each other 
according to constant and universal laws, and give rise in 
one combination to one set of results, in another to another. 
145. It is evident what a revolution these considerations 
cause in the whole view of logic : we see it in the logical 
form of explanatory theory which modern science opposes 
to that of classification, by which antiquity was exclusively 
dominated. I leave it to applied logic to speak of the 
methods which this change in our thoughts necessitates in 
practice, and confine myself to pointing out briefly how the 
logical view of the world, if it were attained as these theories 
understand it, would differ from that of the theory of classifi- 
cation. In the first place, we hear no more of a categorical 
emanation of all real and thinkable matter, proceeding by 
the mere impulse of a plan of development contained in the 
point from which it starts, without the aid of any other 
conditions ; the form of science becomes essentially 
hypothetical. It does not describe what is and wha>: comes 
to be; it defines what must be and come to be //"certain 
conditions are given ; the question whether, and in what 
order and connexion, these conditions occur, is excluded 
from the province of logic and left to be answered by 
experience, which will bring the facts to illustrate the 
application of the theory. Nor will I here raise the question, 
how this theory gets at those universal laws by which it 
decides, that wherever a particular group of conditions is 
given, one particular result and no other must occur ; it is 
sufficient at present to observe that it does start with this 
conception of a law which fixes the particular result of a 


particular condition universally. This means, that wherever 
the condition a 4- b is found, only c follows from it, and the 
nature of the object in which a -f- b is found has no power 
to give this condition directly any other result than c\ it 
can only do so when other conditions, a -j- d, are present in 
it as well as a + , and the former co-operating with the 
latter oblige c to change into y ; and this co-operation also 
takes place by a universal necessity quite independent of 
the nature of the particular object and equally binding upon 
all others. And in the new result y the law which connected 
c with a -j- b is not eliminated, but continues to operate 
concomitantly; for a + d alone would not have produced y, 
but 3. 

From these universal laws arises that mechanical character, 
of which the adherents of these theories make a boast, and 
their logical antagonists a reproach. The tendency to derive 
a series of phenomena ' organically,' as the phrase is, from 
the meaning of a conception which develops itself in them, 
is met by the assertion that a mere meaning which wants to 
develop itself does not produce anything, but that everything 
exists, and exists only, when the complete sum of conditions 
is given from which it follows necessarily by universal laws ; 
it must be regarded as the result of these conditions alone, 
and explanation consists merely in showing that a given and 
perfectly determinate thing is the inevitable consequence of 
the application of universal laws to given and equally de- 
terminate circumstances. Animated by this logical spirit, 
which is found most pronounced in the mechanical sciences, 
explanatory theories are averse both to using and looking for 
universal generic concepts, and to schemes of classification. 
According to them a phenomenon has been merely ob- 
served, not understood, as long as it can be referred only 
to the special characteristics which distinguish one concept 
from others, and not to the prescription of a universal 
authority which is equally binding upon everything thinkable 
and everything real. It is their pride not to need generic 


concepts and their arrangement in a system of classes, but 
to show that, whatever the context from which a phenome- 
non gets its meaning, we know all about it as soon as we 
know the sum of related points combined in it ; for whatever 
is, is merely an example of what must come to be when the 
universal laws are applied to this or that particular group 
of given elements. Even the position which is sometimes 
taken up as the utmost that can be conceded on the other 
side, does not satisfy the demands of these theories, the 
position that everything obeys universal laws, but each 
domain of reality its own, and that the laws of living and 
spiritual existences are different from those of lifeless and 
material ones. It is indeed obvious that those special laws 
to which any given phenomena are immediately subordinate, 
and with which therefore they are most closely connected 
in matter and form, vary with the varieties of the subjects 
which they express ; but there could not be two worlds 
depending on two supreme and independent laws, unless 
they had nothing to do with each other and no effects from 
the one were ever felt within the limits of the other : anyone 
who speaks of one world, embracing those different groups 
of self-developing things and events, must start with a single 
law valid for all reality, or a single unbroken circle of law, 
of which all the special laws of different domains are par- 
ticular cases, and from which they arise as sooft as it is 
supplied, in a succession of minor premisses, with the 
different conditions which differentiate the several domains 
of active existence. 

148. In accordance with my plan of dividing the problems 
of logic, I have omitted from the preceding account of ex- 
planation all mention of the means which the theory employs, 
partly for discovering the universal laws which it assumes 
each coherent group of existence to obey, partly for de- 
tecting in the manifold variety of experience those inner 
coherences themselves which the subordination of different 
elements to the same common principles admits or requires. 


I have reserved to applied logic the utmost freedom to 
follow the course of these efforts ; all that came within our 
systematic survey of the operations of thought, of which we 
are now approaching the conclusion, was the form which 
explanation would like to give to the connexion of all 
thinkable matter, and in which, if it could really be given 
completely, the final goal of intellectual aspiration would 
seem to be attained. As to this goal itself, however, I do 
not share the prevailing conviction of the present day. 
Explanatory theory is almost the only form in which the 
scientific activity of our time exhibits itself; the conscious- 
ness (so late in making itself felt) of the principle which 
that theory has to follow, strongly separates all modern 
science from that of antiquity and the middle ages, and the 
methods of investigation developed in consequence of it 
form the precious treasure which places the modern art of 
discovery far above that of ancient philosophy. Yet the 
opposition so unremittingly made to this form of thought, 
when it claims exclusive dominion over the thinkable world, 
shows that the belief that it leaves nothing more to wish for 
is not universal. If we consider first the familiar forms 
which that opposition assumes in our collective view of the 
world, we shall be able to disengage from it the purely 
logical residuum of feeling which the explanatory theories 
fail to satisfy. 

147. The assertion that all existence is subject only to 
universal laws, and that every individual is nothing more 
than it must become according to those laws, if conditions, 
which might have been combined differently, have as a fact 
combined in a certain form, is most obviously distasteful on 
aesthetic grounds and to artistic natures. Beauty, it is felt, 
cannot be understood upon such a view ; it only seems of 
value, and to be really itself, if the ultimate form which 
excites our admiration is the result of a single power, a result 
which is indeed inevitable, but which, besides being inevit- 


able, is also the fulfilment and manifestation of a living 
impulse : it would appear unintelligible, if it were merely 
a lucky case of harmony between casually coincident 
elements. I have tried elsewhere to show that this 
aesthetic objection is wrong, if it goes on to deny the 
universal validity of the explanatory or mechanical theory. 
As understood by that theory, the meeting of the various 
conditions is never a matter of chance, but always the 
necessary consequence of the past states of the world. If 
we follow out this thought, it leads us back to some com- 
bination of elements which we regard as the initial state of 
the world; and there is then nothing to prevent us from 
supposing that this combination, which might conceivably 
have been different, contained within it the marvellous germ 
of beauty, which, making itself felt through the whole 
mechanical chain of consequences, gives birth by single 
acts of its own to the beauty of single phenomena. Or 
again, if we wish to avoid the difficult conception of an 
initial state, there is no reason why we should not take a 
section, as it were, of the world's course at any point of 
time that we choose, and suppose the combination of all 
the forces then acting simultaneously, just because it is 
that combination and not any other equally conceivable, 
to be the one and sufficient reason of all individual beauties. 
Such a supposition would give room for everything which 
our aesthetic feeling considers necessary to maintain the 
dignity of beauty ; it would merely have somewhat changed 
the place of the single impelling power ; this power would 
no longer lie self-centred in the individual beautiful thing ; 
it would continue to be active in the individual, but only as 
the after-effect of a universal which permeates all indivi- 
dualities. By thus putting back the origin of beauty we do 
not run counter to aesthetic requirements; on the other 
hand, the mechanical theory, obliged as it is to assume 
some existing state of things in which the continuity of 
development according to universal laws is exhibited, has 

Chap. III.] REALITY AND LAW. 1,93 

no motive for conceiving that state as meaningless rather 
than full of meaning, as irrational rather than rational, as 
the source of caprice in the world's course rather than of 
consistent purpose. There is however one point which the 
requirements of aesthetic feeling and the admissions of 
scientific explanation equally imply, namely, that the 
secondary premisses, which we bring under the universal 
laws and by which we denote the facts to which the laws 
apply, cannot have the casual origin which they doubtless 
seem to us to have when we are absorbed in some particular 
field of enquiry and have taken them out of their mutual 
connexion. They must themselves be systematised and 
form parts of a whole, that whole which comprehends all 
real objects to which the universal laws apply. The minor 
premisses to our general view of the world must not be 
conceptions of a number of disconnected possibilities in 
hypothetical form, each of which, if it occurred, would lead 
by universal laws to a definite result ; they ought to dis- 
tinguish categorically each possibility which occurs from 
those which do not occur, and exhibit it as a legitimate 
member with a place of its own in the universal order of 

148. This requirement is partly supported, partly modi- 
fied, by metaphysical considerations. For what would be 
the meaning of assuming on the one side a realm of 
universal laws, and on the other a sum of reality which 
conforms to them, if no further relation existed between 
the two and made this subjection intelligible? And in 
what could the subjection consist if not in the fact that the 
behaviour prescribed by the laws is from the very first an 
actual property of all reality, a constant mark alongside of 
the different or changeable marks by which one real thing 
is distinguished from another ? No truth at any rate can 
be applied^ as we are in the habit of saying, to a given 
content, unless the content itself answers to it ; every appli- 
cation is merely the recognition that what we wish to apply 



is the very nature of that to which it is to be applied. Now 
a limited number of observations enables us to discover 
that everything real exhibits certain constant characteristics, 
and these characteristics then take the shape in our mind 
of expectations which will be confirmed, and which we 
bring with us when we make further observations ; thus we 
easily come to regard them as something which exists inde- 
pendently in fact as well as in our thoughts, and is prior to 
the object in which we shall find fresh confirmation of it ; 
hence all that strange phraseology which regards universal 
laws as powers ruling on their own account, to which every- 
thing real, whatever its origin and whatever its nature, is 
subsequently obliged to submit. If we avoid this wrong 
conception, and connect that which we substitute for it 
with that to which our aesthetic requirements give rise, the 
one and undivided object in which our thought now seeks 
satisfaction is a being, which, not in consequence of a still 
higher law but because it is what it is, is the ground both 
of the universal laws to which it will always conform, and of 
the series of individual realities which will subsequently 
appear to us to submit to those laws. I have no intention 
of exhausting this subject here, and I pass over many 
difficulties which we shall have to notice later, some of them 
in the course of our present logical enquiries, others in 
their metaphysical context : it is enough here to follow out 
the logical form of thought which the mind must look for if 
it tries to satisfy the want just described. 

149. This form will no longer be quite that of inference 
as described above. The universal law, to which the major 
premiss there gave the first place, instead of standing out 
from the other elements as their essential condition, will 
now accompany them as a latent idea, always understood 
but not expressed ; its former place is taken by the universal 
nature of the sum of existence, which is developing itself in 
the world. Nor is this nature conceived as an ideal content 
at rest, which could not be set in motion without extraneous 


conditions, but as the subject of a movement which enters 
into its very constitution and without which it would not be 
what it is. The particular form which the moving content 
assumes at each successive moment, depends on the one 
side upon its permanent purport and permanent direction, 
on the other upon its particular position or the particular 
point to which it has thus far developed, not through ex- 
traneous influences- but through its own movement. It 
would be possible, but would only lead to prolixity, to 
express the essential truth in this kind of idea without im- 
porting into it the conception of motion ; we should then 
find ourselves requiring an idea which includes in the 
system of its species and sub-species the whole of reality ; 
but the differences and the order of these species would 
not be determined independently of the idea by pre-existing 
marks and their modifications ; the idea itself would contain 
the reason for the presence of the marks, for their possible 
divisions, and for the arrangement of the resulting varieties 
according to their value, in fact the whole reason for its 
own classification. We may formulate our requirement 
most shortly as follows : the form of thought for which we 
are looking must have only one major premiss for all its 
conclusions, and this premiss must express the movement 
of the world as a whole ; its minor premisses must not be 
given to jt from elsewhere, but it must produce them from 
itself in the form of necessary and exhaustive varieties of 
its meaning, and thus must evolve in an infinite series of 
conclusions the developed reality which it had conceived 
as a principle capable of development in the major premiss. 
150. It cannot be said that the impulse to organise the 
whole world of thought upon this pattern is foreign to the 
mind when left to itself ; it has been at work at all times, 
and whenever a view of the world more or less like the 
theory of mechanical explanation has developed itself, this 
impulse has met it with the reiterated demand that the 
world and all things in it should be regarded as a living 

O 2 


development. For it is in the phenomenon of life that we 
believe ourselves to see these claims of the mind com- 
pletely satisfied ; as there the original type of the organism 
is made into the efficient power which produces the incen- 
tives and conditions for its own consistent development, so 
we would have the world as a whole evolve from itself the 
occasions which are the necessary conditions of its gradual 
self-realisation. We need not here notice the errors in this 
belief in the independent development of the individual 
organism; it is enough that it appears to be a graphic 
instance of what we are looking for. The same image has 
also been a constant favorite with the theory which, for 
the last time in our day, avowedly aspired to a vision of the 
universe springing out of the unity of an idea, which 
develops itself and creates the conditions of its progress. 
For it was in no attitude of investigation and reflexion, by 
no means of logical and discursive thinking, bringing in- 
dependent minor premisses under universal majors, that 
the Hegelian philosophy even wished to derive the world 
from its single principle : it only proposed to look on and 
see how the development followed from the inherent impulse 
of the idea. And for this intellectual vision, this ' specu- 
lathe"* thinking in the original sense of the word, it believed 
itself to have found a guide in the dialectical method, a 
guide which enables the spectator to follow the true course 
of the self-realising development. I shall still keep to my 
principle of saying nothing in this survey of logical forms 
about the practical rules for securing their application to the 
matter of thought, and therefore leave for a later occasion 
what is to be said about this method as a method ; but I 
shall appropriate the antithesis between speculation and 
explanatory theory for the purpose of describing the final 
shape which we aim at giving to all thinkable matter, and 
call the form of speculative thought this third member, with 
which the series of comprehensive and systematic forms 
comes to an end. 


151. And yet I feel that I must not conclude quite so 
shortly ; I must return once more to an observation which 
I have already made. All forms of thought which we are 
considering are ideals ; they indicate the final shapes which 
thought wishes to give, or to be able to give, to the matter, 
great or small, which it has before it, in order to satisfy its 
own inherent impulse by showing the coherence of all that 
coexists. Nor is the validity of these ideals at all impaired 
by the fact that human knowledge is not able to apply them 
to every given instance. It may be that we are not always 
in a position to discover the universal laws which govern a 
particular circle of phenomena ; and it may be that, if we 
had discovered them, we should not succeed in bringing all 
particular cases under them so completely that the necessity 
of any given result was at once apparent. But we should 
not push forward our enquiries in this direction so untiringly, 
if we were not convinced that the principle of the explana- 
tory theory is universally valid, and that its validity is 
independent of our present ability to verify it in every 
conceivable instance. Perhaps the form of speculative 
thought is in a still more unfavorable position ; the con- 
ditions under which human thought is placed may be 
altogether inadequate to achieve the speculative ideal in 
more than a few instances, perhaps even in one ; yet this 
ideal also- will retain its binding force, and continue to 
express the form in which, if we could give it to the whole 
material of thought, our mind would find all its demands 
satisfied. This form also, therefore, has a right to its place 
in the systematic series of forms of thought : that it is the 
last in the series is clear without proof, for it leaves no 
elements remaining in mere unconnected juxtaposition, but 
exhibits everything in that coherence which had been all 
along the aim of thought. At the same time it points 
beyond the province of logic. From the point of view 
of the explanatory theory it might still seem as though the 
universal laws, which thought produces from itself alone, 


gave a right to decide a priori what reality will be like ; 
speculation does not deny this right, but by making the 
content of a supreme principle the one and only ultimate 
ground of everything, both of the power of these universal 
laws themselves, of the direction in which the world as a 
whole develops, and of the individual forms which in con- 
sequence reality assumes at each moment, it indicates that 
the final fulfilment of all logical aspiration could not be 
attained by new logical forms, but only by material knowledge 
of that supreme self-developing principle which speculation 

In concluding this account I am conscious how much its 
method deviates from those which are in vogue at the 
present day. We are so accustomed to being told the 
history of things, and to feel our curiosity satisfied when we 
have discovered or invented an origin for them, that even 
logic is flooded with psychological explanations and deriva- 
tions of its doctrines : on the other hand it strikes us as 
antiquated, odd, and unmeaning if anyone attempts to 
arrange the forms of thought in a progressive series 
according to the nature of its problems, instead of following 
the order in which the mental activities necessary to their 
solution develop in the individual soul. I am content that 
this should be so, and hope that in the form of my ex- 
position my readers will recognise the premonitory influence 
of the idealistic philosophy to which it is intended to lead : 
I have no fear that by choosing this form I have distorted 
the substance of truths which, on any view of logic, must be 
equally regarded as established. 




152. WE are so much accustomed to oppose the world 
of our thoughts to an external reality, that as soon as we 
speak of an object to which the forms of our thinking are 
to be applied, it seems as if we can mean thereby nothing 
but this external reality. When we call to mind the natural 
sciences, which occupy so large a portion of the field of 
science at the present day, we are confirmed in this opinion ; 
on the other hand, when we think of mathematics and 
jurisprudence we are likely to be shaken. The external 
reality supplies neither the objects with which the mathe- 
matician^ deals nor the methods by which he deals with 
them. That which it yields does but give him an occasion 
to turn his investigations in this or that direction. The 
true objects of his enquiry are always nothing but the forms 
which our intuition or our thinking finds in itself or creates, 
and of which the appearances of the outer world remind us, 
without ever perfectly corresponding to them. And his 
business is, in accordance with laws of reasoning, which 
at any rate are not derived from any external experience, 
to develop the countless necessary conclusions which follow 
from the various possible combinations of these forms. 
Nor is this development speedily achieved : these con- 


sequences do not unfold themselves in such a way that 
we need but to look on and watch : on the contrary logic 
has at all times turned to mathematics (for the two are 
coeval) for examples of delicate profound and fruitful 
methods of enquiry. 

Jurisprudence certainly owes the occasion of its origin 
to the circumstances of the actual world in which man with 
his needs and claims is placed; but it tries to shape this 
world and our relations to it by ordinances, which, though 
as against nature they are products of our free choice, 
are yet the necessary consequences of ideas of right and 
justice, consequences of a truth that ought to be, which 
has its home nowhere but in our own minds. And so 
logical acumen is just as constantly employed here also 
in setting forth ever more precisely and irrefragably the 
connexion of the several conclusions already drawn both 
with one another and with the highest principles from which 
they flow. 

Thus both these branches of science show that logic 
need not go to the external reality to find objects for its 
application, that it finds fully work enough in investigating 
the connexion of that which is possible in thought and 
necessary in thought, that finally the inner world of our 
conceptions is wide enough to contain unknown regions, 
still to be discovered by means of systematic enquiry. 

153. Keeping to this line of thought we may now turn to 
the natural sciences. Even the external world which we 
assume is after all an object of our enquiry only so far 
as (in some way or other which does not here concern 
us) it has become a world of conceptions in us ; we survey, 
dissect, and investigate not that invisible something which 
we suppose to lie outside us, but the visible picture of 
it that is formed in our consciousness. We may believe 
that we are compelled, as the result of prolonged labour, 
to accept certain connexions according to law between the 
unknown parts of this unknown external something ; but 


all these assertions (whatever they may be) are after all 
grounded solely upon the relations which prevail either 
persistently or in succession between the contents of our 
thoughts. Whatever may be the causes which produce 
this succession, the laws by which it is regulated can 
only be known by itself, i. e. by the order in which certain 
thoughts follow certain others in our minds, by the constant 
union of some thoughts, and the impossibility of uniting 
others. It is enough then even for the treatment of the 
external world to regard it in the first instance as a world of 
thought set up somehow or other in us ; whether the ap- 
pearances which surround us correspond to a real world 
of external things, or whether they be products of a creative 
faculty of imagination in us, guided by unknown impulses, 
the discovery of the connexion between them will always 
necessitate the same methods of enquiry. 

I wish the reader to bear in mind what I have said 
as we pass to applied logic. My purpose in saying it here 
is only to indicate the position taken up in the following 
enquiries : in the course of these enquiries we do no 
violence to the ordinary way of thinking; let the reader 
while he reads these chapters conceive of the efforts of 
thought as directed to a real external world ; only when 
he finds no notice yet taken of the relation of this world 
to our thought, I hope he will find a justification of this 
course in these few prefatory remarks, and be content 
to wait till the third part of my treatise for an enquiry 
into the significance of the issue which is here put aside. 


The forms of Definition. 

154. INNER states, sensations and ideas, feelings and 
impulses, cannot be conveyed like material things, which 
may be separated from their original possessor and passed 
on as they are from hand to hand. We can communicate 
them only by subjecting our neighbour to conditions under 
which he will be compelled to experience them or to beget 
them anew in himself. 

If we had to communicate for the first time something 
yet unknown, which was too simple to be created by 
thinking, or too complex to be exhausted by it, our only 
resource would be to produce the external conditions of 
perception. If our neighbour had never seen light, or 
heard sounds, or felt bodily pain, our only course would 
be to put his eye within reach of a source of light, to bring 
waves of sound to act upon his ear, and by the application 
of a stimulus to his body to let him experience that feeling 
of pain with which we ourselves had made acquaintance 
in precisely the same way. If we wish to enable him to 
recognise a person whom he as yet does not know, the 
description of the countless little marks which distinguish 
that person from others will never make sure, but by 
pointing with the finger we can show him precisely whom 
we mean. We need do no more than thus barely mention 
the fact that wherever it is applicable this direct reference 


to the object itself or to some likeness of it is always 
useful. But in view of the questions which here concern 
us we further presuppose two things, first a large stock 
of past experiences common to the persons who are to 
communicate with each other, and secondly a language 
intelligible to both parties, to the several words of which 
each attaches (to a large extent at least) the same ideas. 
Then by a series of spoken words we call to our neighbour's 
recollection the ideas conjoined with them in that order 
which is for him the internal condition of his creating or 
experiencing in his own consciousness that which we wish 
to communicate. 

155. This form of communication also includes much 
else that our logical enquiry can only take note of by the 
way. Both poetry and eloquence aim by this method at 
something more than imparting ideas : they count upon the 
attachment to the images thus called up of feelings of 
pleasure and pain, of approval and disapproval, of exaltation 
and aversion. The effects which they thus produce are 
powerful but uncertain. Different minds are indeed pretty 
uniformly organised for the mere apprehension of matters 
of fact, and their general habits of perception do not 
change; but in estimating the degrees of emotion which 
we annex to what we perceive we must allow not only 
for original differences of temperament, but also for 
the changefulness of the mood of the moment, which 
depends upon what we have just gone through. Thus 
different persons are very differently receptive even of actual 
facts ; still less can we hope by the imperfect recollection 
of such facts, which is all that speech can rouse, to create 
in others precisely the same emotion which they produced 
in ourselves. How much may be done by skilful guidance 
of the train of ideas and by well-measured expressions to 
lessen the uncertainty of the result is a question for the art 
of poetry and rhetoric. Our own problem is narrower and 
is limited to the communication of that which has been 


already refined from a state in which we are acted upon 
into an idea which we apprehend, i.e. of thoughts, not of 
feelings and moods. 

156. The certainty even of this kind of communication 
seems to be imperilled by the fact that after all the same 
words do not always have the same meaning for the speaker 
and the hearer. It must be allowed that, apart from 
subsequent confusion of originally different roots, there are 
in every language many words which denote several very 
different things, in consequence no doubt of a resemblance 
which these things bear to one another, but still of a 
resemblance which is not always so obvious now to him 
who uses the traditional words as it was to the first inventor 
of these metaphorical expressions. And even when a word 
denotes the same thing for all, that does not ensure that all 
have the same conception of the thing denoted. The 
special circumstances under which each individual became 
acquainted with the thing, the peculiar point of view from 
which he first regarded it, the connexion in which he found 
it and from which he had to detach it, give a peculiar 
colouring to his picture of it, and dispose him to other 
conclusions than those anticipated by the speaker when he 
named the common word, hoping thereby to give some 
particular turn to the course of his hearer's thoughts. It is 
impossible to deny these facts, dangerous to disregard them 
altogether, yet foolish to press them too far : the intercourse 
of daily life sufficiently proves to how large an extent speech 
enables us in spite of them perfectly to understand each 
other's thoughts about the most various matters. There 
will certainly remain ideas which it is hard to communicate 
with precision ; but were there no such difficulties there 
would be no good in seeking rules for helping us by the 
appropriate use of unequivocal words to remove the ambi- 
guity of others and to fix their meaning so that all who 
wish to converse may use them in the same sense. It 
must be left to the unfettered acumen of the speaker to 


determine what words may be accepted as precise enough 
to explain other words ; but however far we may feel con- 
strained to go back along this line and to remove all 
ambiguity from the instruments of communication which 
we wish to use before we use them, there will still be only 
two possible ways for us, abstraction and construction. 

157. We explain a conception, which we will call M 9 by 
abstraction, when we first refer to a number of known 
instances, in each of which M forms a part of the notion, 
and then bid the hearer separate from these instances that 
which does not belong to the conception M which we wish 
to communicate. This is the way in which all our general 
conceptions 1 and general ideas 2 were originally formed; in 
the case of a general idea that which was common to a 
number of impressions comes of itself t stand out as the 
object of a new separate idea ; in the case of a general 
conception this process is consciously directed by attention 
and reflexion. And when we are at a loss we all come 
back to this same way. The man of no logical training 
does so when to the question what he understands by M he 
replies, in the fashion which the Platonic Socrates so often 
complains of, only by giving examples which contain M y 
leaving to his questioner the trouble of separating the 
common element which he wants to get at from that which 
is foreign to it. But the logically trained thinker also 
proceeds really in the same way: however carefully he may 
choose his terms so as to express the universal itself with- 
out any reference to particular instances, yet this expression 
is only obtained by a tacit comparison of a number of cases. 
It is only by such a comparison that we learn what marks 
of M must be precisely fixed in order that the expression 
may exclude all that is foreign to M, what other marks 
must be left undetermined in order to include in M every- 
thing that is properly an instance of it. And lastly, only 
by the fact that instances are to be found are we convinced 
i [' Begriffe.'] a [' Vorstellungen.'] 



that this My which we are taking the trouble to determine, 
is capable of determination, that it represents a problem 
which has an intelligible solution, not a mere tissue of in- 
compatible elements whose union may be demanded in 
words but cannot be really carried out. 

158. It is thus useful to follow this method of abstraction 
in every case, and even when we may have arrived at a 
determinate conception in some - other way, at any rate to 
confirm it by a supplementary reference to instances. 
Wherever our aim is to fix some very simple conception 
which underlies a whole group of kindred ideas, it is the 
only method possible. Such a conception can only be 
pointed out by taking away from known instances of it all 
that does not belong to it ; we can never put it together 
out of its component parts, for it has none. The labour 
expended upon this impossible aim always ends in a vicious 
circle, since among the materials that are to be used in the 
construction the very thing that was to be constructed is 
taken for granted, whole and entire, however much it may 
be concealed under strange expressions. Thus, for example, 
in our idea of becoming the two ideas of being and not-being 
are no doubt united as two connected points of relation ; 
but if we should try to characterise becoming as the unity 
of the two we should not attain our object. In the first 
place we should be bound to fix the precise sense to be 
here assigned to the expression ' unity ' which in itself is 
very ambiguous. It cannot mean the mere co-existence in 
the same consciousness of the two ideas of being and not- 
being, for obviously becoming is the content of a relation 
that exists between the contents of these two ideas. But if 
we try to unite being and not-being as predicates applicable 
at the same time and in the same manner to one and the 
same thing, we do not arrive at becoming, but simply find 
ourselves confronted by the impossibility of actually exe- 
cuting in thought a task which involves such a contradiction. 
Suppose then that we separate again the being and the not- 


being of this thing and say that the one predicate is applic- 
able to it when the other is not : even by this change we 
do not get hold of becoming; it falls between the two 
moments of time and is to be found in neither. We shall 
have therefore to bring them together once more : but as 
long as they are separate from one another becoming will 
lie outside of them, we can only get hold of it when we 
look for it neither in being nor in not-being, nor in a passive 
unity of the two, but in the transition from one to the other. 
But in this idea of transition, or in any idea however it be 
expressed that we like to substitute for it, we shall recognise 
(only under another title) what is essentially our idea of 
becoming. This relation therefore between being and not- 
being, as it is altogether sui generis, cannot be conceived by 
means of anything but itself, is only to-be got by abstrac- 
tion from the instances in which it forms a part of the 
thought, not to be created by the putting together of ideas 
which as yet do not contain it. Precisely the same con- 
siderations hold with respect to the equally simple concep- 
tions of being, acting, thinking, affirming, denying ; and the 
geometry of Euclid follows precisely the same method in 
determining the surface as the limit of the space occupied 
by a body, the line as the limit of the surface, the point as 
the limit of the line, in each case teaching the learner to 
get the simpler conception, which is harder to grasp, by 
abstracting what does not belong to it from the more com- 
plex conception which lies nearer to sense or which has just 
been determined. 

159. The opposite method would fully deserve the name 
of construction only if it enabled us completely to put to- 
gether the idea to be conveyed out of a definite number of 
unequivocal parts by a series of acts of thought which we 
were required in unambiguous language to execute upon 
those parts. Almost the only conceptions that really admit 
of this treatment are the mathematical conceptions and 
some others that arise out of the applications of mathe- 


matics, conceptions which as creations of our thought 
contain only what our thought has combined in them. 
They admit of it because the several ideas which make up 
the whole conception can be completely enumerated, and 
because not only each of these ideas but each of the ways 
in which they are to be joined together is such that we can 
state the characteristic quantity by which it is distinguish- 
able from others of its kind, as well as the special quality 
which distinguishes it from those of another kind. Here 
then nothing remains indeterminate that should be deter- 
mined ; he who follows the directions given must see the 
picture he is desired to form rise before his mind's eye with 
just that degree of individuality or generality which the 
speaker wished to give it. 

If on the other hand we wish to convey a notion of some 
really existing thing we are met by well-known difficulties. 
Our mental picture of a real thing is not made up of a 
limited number of points of relation which are to be brought 
into combinations also limited in number, but is com- 
pounded of a countless number of ideas. And of these 
component ideas those that belong to different senses can- 
not be compared with one another, while even those of the 
same sense can only be designated by general names, and 
scarcely admit of precise measurement. And lastly it is 
beyond our power to make a complete survey of ..the com- 
binations of all these elements, nay we cannot perceive 
them at all except so far as they consist of an external 
arrangement in Space and Time, and even then we cannot 
find any comprehensive expression for them in our ignor- 
ance of any pervading law of their formation. 

In the presence of this fulness of detail construction 
shrinks into description. In describing we try, if we under- 
stand our business, first to fix the main outlines of the 
whole idea, whether this be done by a simple construction, 
or by taking as illustrations similar things already known 
and proceeding by alteration and transposition, by the 


removal of some features and the addition of others, to 
elicit from them the leading lines of the picture we wish to 
convey. Then we fill in the mass of details, never com- 
pletely, for they arc usually inexhaustible, but skilfully 
selecting those by the mention of which we may hope that 
the hearer's attention will be at once stimulated to supply 
from his own memory those that are not mentioned. We 
need but remind the reader of the wonderful effects which 
the poet produces in this manner, bringing a whole picture 
before us with a touch ; though the uncertainty of the result 
is equally manifest. The way in which each man supplies 
what is not mentioned vanes according to his nature : were 
it possible to bring to view in detail the different pictures 
which the same description calls up in different hearers, 
their variations would show what an inadequate basis a 
description must be for the support of definite conclusions. 
For scientific purposes therefore description needs a regula- 
tion of its method, and this it finds in the rules of defini- 

16O. For the definition of a conception M it is usual to 
require a statement of the next higher generic conception 
G (the genus proximum\ and of the characteristic mark d 
(the differentia specified) by which M is distinguished from 
other kinds of G. By requiring the generic conception G 
we set bounds to the arbitrary and capricious course of 
description. In describing you were free to begin at any 
point whatever, and then gradually to add the remaining 
points in any line that you pleased, so long as you could be 
sure of producing in the end a clear picture of what you 
meant But even in a description you would not attain 
your end without the employment of many general con- 
ceptions. Now instead of an arbitrary choice of these, the 
rules of definition require you to start from that universal 
conception in which the largest part of the constructive 
work before you lies completed and ready to hand, and 
which, being denoted in speech by an unequivocal name, 



may be assumed to be familiar to every mind, fitted to serve 
as the outline for the filling in of the details by which the 
intended picture is completed. 

If we are told that a creature we have never yet seen is a 
bird, this general conception gives us at once a clear picture 
of a number of members united in a characteristic manner, 
and at the same time of the peculiar kind of locomotion 
and vital action to which they are instrumental. The 
further special characteristics are easily added to this out- 
line, for it indicates of itself the places to which they 
severally belong. We should never get such a clear idea of 
the unknown creature if we had to put it together out of its 
primary components. It would be an endless task to 
enumerate all the variously-coloured spots on its body with 
their position and the extent to which they may be dis- 
placed, so as to give a notion even of what it looks like. 
Still more endless would it be to add to this the peculiarities 
of life and habit, which all belong at any rate to our idea of 
the animal in question if not strictly to our mental picture 
of it. 

We see then the value of the abbreviation effected by 
starting from a general conception that can be assumed as 
known : we understand also that we must choose for start- 
ing-point not merely any higher universal, but expressly the 
genus proximum, which in its characteristics and in the 
mode of their combination comes closest to the conception 
to be defined, and so clearly describes the point at which 
and the manner in which we are to add each of the last 
characteristics by which the conception is finally determined. 
By starting from a higher universal than this we should not 
only lengthen again the rest of our task, which definition 
was intended to shorten, but we should run a risk of failure. 
For we should then have to add a whole series of further 
characteristics in order to exclude everything foreign in the 
long descent from that less determinate universal to the 
particular species in question : and each new characteristic 


would open a new source of error ; for it is hardly possible 
to determine quite precisely the mode and manner in which 
each is to be added to those that have preceded it without 
appealing to a picture which it may be assumed that each 
man already has in his mind. The notion of that genus 
proximum therefore would not by this method be produced 
afresh with that definiteness and certainty with which it 
could be recalled to the memory at once by the mention of 
its name, and which it must have if it is to serve as an out- 
line for the filling in of the final characteristics of the con- 
ception which we desire to convey. All that we could get 
by this method would be more or less of a riddle. For 
when we propound a riddle what we do is this, we tell our 
hearers without more ado to attach to a very indefinite 
universal (a mere something that may be anything) predi- 
cates that can be united only in one very definite subject, 
leaving it to his ingenuity to find this subject or in the first 
instance the genus proximum which admits of their union. 

161. As yet we have spoken of the definition as a me- 
thodical description. If it is to retain this character it would 
have with regard to M to state completely the modified 
forms p 1 <? 1 r l assumed in the case of M by P Q J? the 
general predicates of the genus G. Instead of all these 
characteristics the usual rule for definition requires us to set 
down on!y one characteristic d, the specific difference, by 
which M is distinguished from all other species of the 
genus G, Definition thus has a more limited and therefore 
a more practicable aim than description : instead of setting 
forth positively the whole content of M it has only to state 
the mark by which M may be separated from all that is not 
M. This is the origin of the terms dcfinitio and opines, 
both of which imply only the marking off of one thing from 
another. And in fact the general aim of definition must be 
thus limited. As thought advances we feel no doubt the 
need not only to distinguish, but to know completely what 
we have distinguished.; then we make further demands 

p 2 


upon definition ; then we refuse to admit as a specific 
difference anything but one of those characteristics that 
really make a species, i. c. one whose occurrence decisively 
modifies the forms assumed in M (the thing to be defined) 
by all the other characteristics of the genus G which are not 
mentioned in the definition. These heavy demands how- 
ever can be completely satisfied only at the conclusion of an 
enquiry which has made us perfectly acquainted with the 
nature of Jlf, and which thus enables us to solve the problem 
which remains, of fixing a final and classical expression for 
that nature. 

But besides this there are other no less pressing problems. 
We may have to begin a speculative enquiry, which has to 
find a number of yet unknown propositions that are true 
of M } or in a practical matter we may have to determine 
what is the proper consequence of a given situation M: in 
either case it is of the utmost importance that this M, to 
which the propositions we are going to assert or the decision 
we are going to arrive at must apply, should be marked off 
by precise and easily traceable boundaries, nay at first this 
is the only thing that is of importance. For this purpose 
any characteristic d will suffice, even the most insignificant, 
provided only that it be really an exclusive mark of M. In 
the first case, that of a speculative enquiry, the further 
course of the enquiry itself will either reveal the reason 
which connects the validity of a series of propositions with 
the presence of this obscure characteristic d, or will show 
that they are valid over a wider or narrower field than this, 
so that d is not the proper characteristic of their subject. In 
the other case, that of a practical matter, the exact meaning 
of a legal situation to which a law is to apply must be 
completely considered beforehand while the question is still 
de legj ferenda ; but he who has to carry out the lex lata 
rightly demands that this previous consideration shall have 
given the law the form of a definition which distinguishes, 
not by the most profound but by the most obvious mark, 


the cases to which a decision shall apply from those to 
which it shall not. These are problems which applied logic 
cannot decline, and we overlook them when we think too 
disparagingly of this traditional form of definition. We 
misunderstand the sound sense of many such definitions in, 
practical philosophy and jurisprudence when instead of the 
marks of M, which they intend to give and do give com- 
pletely, we see in them nothing but an inadequate statement 
of the whole nature of -#/, which it is not their purpose to 
give at all. 

162. It will be convenient to notice in this context the 
distinction which is commonly drawn, but not always in the 
same sense, between nominal and real definitions. We may 
utter a name or replace it by another; but we can never 
define anything but its meaning, i.e. our idea of that which 
it is intended to signify: the thing itself again is not in our 
mind, but only the picture we have formed of it. These 
two kinds of definition therefore seem to be identical ; and 
they are in fact identical for everything that exists only in 
our minds, and whose whole nature therefore is exhausted 
by our idea of it. There is no real definition of a geometrical 
figure that can be distinguished from its nominal definition ; 
any correct definition that we give of it expresses at once 
the whole nature of the thing in question, and the whole 
meaning of the name. 

In other cases however the distinction between these two 
modes of definition is one that it is worth while to make. 
If we call the soul the subject of consciousness, of thinking, 
feeling, and willing, this may be appropriately called a 
nominal definition ; it specifies a condition which a real 
thing must satisfy if it is to be entitled to the name of a 
soul. But who or what this thing is whose peculiar nature 
enables it to satisfy this condition, is still quite an open 
question ; we have not fixed the real definition of the soul 
till we have got a theory which proves either that only a 
supersensuous and indivisible being, or that only a con- 



nected system of material elements can be the vehicle of 
consciousness and its various manifestations. It was a 
nominal definition of beauty that Kant gave when he said 
that it is to be found not in the conformity of the beautiful 
object with some conception, not in its capacity to satisfy a 
desire in us, but in the fact that it pleases directly and 
without reference to any interest. The real definition of 
beauty would have to point out the precise relations between 
various things or components which enable every object in 
which they occur to produce this pleasing effect. And so 
we may say in general terms, when experience shows us a 
group of characteristics p qr often occurring and continuing 
together, or when in the course of our investigations we 
light upon a coincidence which induces us to put them 
together and to regard the group as a subject for further 
enquiry, we proceed in the first instance to form for the 
group a conception M, of which a nominal definition can 
always be given, because it has only to set forth the predicates 
which led us to invent the name, or the effects which we 
expect from the thing to which the name is applied. But a 
real definition cannot always be given : for there is no 
assurance that we have not combined in M characteristics 
whose union we thought ourselves justified for some reason 
or other in assuming or desiring, when there is in fact 
nothing to be found in which they really are or can be 
united. It is a common error to mistake this mere indica- 
tion of a problem we should like to solve for the solution 
itself; and on this account the distinction between these 
two kinds of definition is useful as a warning. 

163. We have to beware of three faults which vitiate 
a definition. 

In the first place its assertion M=Zmust be no tautology ; 
but it becomes a tautology whenever M itself is explicitly 
or implicitly assumed among the ideas combined in Z by 
which M is to be explained. This fault (called drculus in 
definiendo) is often committed through carelessness which 

Chap. I.] FA UL TS OF DEflNl TION. 2 1 5 

no rules can prevent ; but we are almost of necessity driven 
to it whenever we try to give a formal definition of 
some simple thing which does not fall under any more 
general conception. 

In the second place a definition, since it has to fix a 
conception, must be a universal proposition, true of every- 
thing which falls under the conception. Now if every 
J/=Z, it follows by contraposition, that no M is not-Z: 
if then further reflexion or fresh experience teaches us 
that after all there are some M which are not-Z, we know 
that the definition M~Z was too narrow (definiendo 
angustior) and was not, as it ought to have been, true of 
every M. 

Lastly a definition must be convertible : if every MZ, 
it must also be true that every Z J/: whenever therefore 
further reflexion or fresh experience shows that some Z 
are not M, we know that the definition MZ was too 
wide (definiendo latior)^ and included some non-J/ which 
it ought to have excluded. 

To point out how to avoid these faults would be more 
useful than thus merely to name them ; all we can do 
in that way however is to indicate their usual source, 
viz. the limited range of our observation, which as a rule 
opens to each individual only one and the same fragment 
of the entire field covered by a conception, and further the 
one-sidedness into which our thinking is apt to lapse if 
it does not constantly receive fresh stimulus from without. 
In the temperate zone the way in which plants awake in 
summer and sleep in winter makes a strong impression 
upon our feelings ; animal life, with its continuous activity, 
seems to offer a complete contrast. Now we certainly 
should not base upon this a scientific distinction between 
animal and plant ; yet countless comparisons, employed 
by poet and orator, show that we are accustomed to con- 
sider this yearly alternation as the essential characteristic 
of the plant. But a definition which expressed this would 


be at once too narrow and too wide : it would exclude 
tropical plants whose life is an uninterrupted growth, and 
would include hibernating animals, which in this climate 
easily escape our attention, directed as that is mainly to 
the domestic animals. It may easily happen that one who 
wishes to establish on a new basis the rights and duties, 
both political and social, of all the members of the state, 
thinks only of the male world to which the conduct of 
these transactions is usually confined, and then his pro- 
posals will be too wide, in as much as he demands for 
all what he intends for men only, or too narrow, in as 
much as he expressly enacts for men only what must 
obviously apply to all. From this we may draw a lesson 
of universal application : we should never attempt to treat 
a problem off-hand, when it is possible to extend the 
limits of our own experience by converse with others or 
by taking count of views which are already recorded in 
the literature of the subject. Learning is not in itself 
inventive, but like any other training and discipline, it 
makes us more secure against extreme errors than if we 
proceed by the mere light of nature. 

164. We further require in a definition elegance and 
brevity, which I will illustrate by a simple instance. If 
we define a circle as a curved line all the points of which 
are equidistant from its centre, we first of all make an 
actual mistake in giving too wide a definition. For if on 
the surface of a sphere we draw a serpentine line which 
crosses and recrosses a great circle of the sphere making 
equal curves on either side, all the points of this line 
are equidistant from the centre of the sphere. 

If further the line, in returning to its origin in the great 
circle, describes an uneven number of these double curves, 
it will consist of an infinite number of pairs of points, form- 
ing the opposite, extremities of so many diameters of the 
sphere. The centre of the sphere therefore bisects the rec- 
tilinear distance between the two points of each pair ; and 



so, in every sense which can here be given to the word, it 
would also be the centre of the sum of all these pairs, i.e. 
of this line, which nevertheless would not be a circle. We 
ought therefore to have said that a circle is a curved line in 
one plane which fulfils the above condition. 

But elegance further demands that a definition shall not 
contain more ideas than are indispensable for the complete 
determination of the given conception. So we may be 
called upon to speak not of a curved line but of a line 
simply: if a line fulfils the annexed condition it follows 
without more ado that it cannot be straight. The condition 
itself however is not correctly expressed. A definition 
should not employ among its instruments of explanation 
ideas which are themselves unintelligible without the con- 
ception to be defined. In this case the idea of the centre 
is certainly such an idea. If we had not yet got the idea of 
a circle (and in fact there is nothing in this case at least to 
suggest this idea to us, after we have omitted the character- 
istic of curvature from our definition) we could at first think 
of the centre of a line only as the point of bisection, and 
we should not discover our error till w r e attempted to con- 
struct a circle on that understanding. Instead therefore of 
this sense of the term centre which common usage suggests, 
and which compelled us to be so painfully discursive just 
now in speaking of our serpentine line, the definition re- 
quires the precise statement in general terms of the meaning 
which the word is to bear for all figures whatsoever. This 
statement can easily be given, but I may omit it, as it 
follows therefrom that //there be a point in a plane which 
is equidistant from all the points of a line in that plane, that 
point is the centre of the line. But if we now introduce 
this definition of centre into our definition of a circle, the 
statement of the further condition under which the line in 
one plane becomes a circle comes to be a mere tautology, 
and the meaning of the whole definition is evidently nothing 
more than that a circle is a line in one plane for which 


there is a point in the same plane from which all its points 
are equidistant. The definition is substantially correct ; 
yet fault may be found with its form. For now after 
omitting the term centre we remember that it was only the 
presence of that term that forced us to look for the equi- 
distant point, in the same plane. Not this actual centre 
only, but any point in an axis drawn through it at right 
angles to the plane of the line fulfils the condition of being 
equidistant from all points of the line. It is enough there- 
fore to say that a circle is a line in one plane such that a 
point may be found from which all its points are equidistant. 
It is needless to mention that there are several such points 
and to say where they lie : the attempt to construct the line 
according to this direction will at once teach us both. But 
once more even in this form the definition is not quite all 
that can be desired. It does indeed say that all the points 
of a circle are equidistant from one and the same point, but 
it does not formally state whether or no all points that are 
equidistant from this point are points in the circle. They 
are so in fact provided they he in the same plane, and thus 
in order to express this along with the rest we may finally 
say that a circle is a line which contains all the points in one 
plane which are equidistant from any point. 

165. Different opinions may be entertained as to the re- 
quirements of definition which I have just illustrated by the 
example of the circle. Every one will allow that it is a 
serious fault to employ ideas which (like centre in this case) 
though a meaning may be given them apart from the con- 
ception to be defined, yet are not fully intelligible without 
it, except perhaps in the context of a scientific treatise. But 
it may be thought that the addition of superfluous cha- 
racteristics is unobjectionable, since it makes the definition 
easier to understand without impairing its correctness. 
Nevertheless it should be avoided. For the addition of 
some characteristic z that might be dispensed with, is apt, 
as we are not told that we might dispense with it, to make 


us think that it is inserted in order to distinguish the M we 
are defining from a non--#/to which everything in the de- 
finition is applicable excepting only z. If we say a circle is 
a curved line in one plane such that there is a point from 
which all its points are equidistant, the form of the state- 
ment suggests that there are also straight lines which satisfy 
that condition. It matters little in so simple a case as this ; 
but in more complex cases serious disadvantage may be the 
result of this apparently harmless addition of superfluous 
matter. At the least it hampers us in the drawing of con- 
clusions, which after all was our sole purpose in laying down 
the definition. It may happen, for instance, that it has 
been quite clearly established, perhaps in some indirect 
way, that Q has the whole sum of predicates that are suffi- 
cient according to the correct definition for the subsumption 
of Q under M, but that it is difficult or impossible to prove 
directly that Q also has the predicate z which is superfluously 
added in the definition actually given : there will then be a 
quite useless hesitation about bringing Q under M and 
actually drawing the conclusion which that would justify. 
And so we may say generally that it is right to demand 
that a definition shall contain only those terms that are 
indispensable for the specification of the object, but shall 
exclude all merely descriptive elements : if it does not 
enable us very readily to form a picture of the thing, this 
will be atoned for by the certainty of the conclusions we 
can draw from it. 

166. Hitherto we have been considering the usual form 
of definition by the proximate genus and the specific 
difference as the only valid form. But the untrained in- 
tellect is wont, to the annoyance of the logicians, to use 
another mode of definition, and to say, for instance, in its 
familiar uncouth way, sickness is when something pains me. 
Such a phrase certainly needs to be amended, yet not 
exactly in the way which logicians rather intolerantly 
require, but rather in the way in which physical science 


actually defines many of its conceptions. The ordinary 
form is properly adapted only for defining the meaning of 
a substantive : when we have to do with adjectives and 
verbs it is not only shorter but more correct to give them 
their proper place in the grammatical structure of the 
definition, and to let them bear plain reference to their 
subject, seeing that it is only as expressing states or pro- 
perties of a subject that they have any meaning. It is quite 
right therefore to define adjectives like sick or elastic by 
such propositions as 'a living organism is sick when its 
functions depart from a certain course;' 'a body which on 
the cessation of external constraint resumes its original 
shape is elastic. 1 And in defining the meanings of the 
verbs to live and to sin it would be quite proper to name 
first the subjects to which they can be applied, an organic 
body and a spirit that is conscious and wills, and then the 
conditions under which they are to be predicated of these 
subjects. It is absolutely useless to begin by throwing all 
these ideas into the substantive form and ranking them 
under the head of states or properties or modes of action : 
that they are to be so ranked is at once apparent if we 
leave them their adjectival or verbal form and give them 
their proper place in the sentence. The usual mode of 
definition on the contrary has the disadvantage of making 
us far too apt to separate from its subject and treat as 
independent what is nothing but a state or property of 
something else. When we have once framed the substan- 
tives sickness, sin, freedom, it is hard to keep quite clear 
of the strange mythology which speaks as if these terms 
stood for things with a being of their own, and traces their 
development, without ever seriously coming back in the 
course of its enquiry to their real subjects, though it is only 
as properties, states, or activities of these that they exist, 
and though their apparent development is every moment 
bound up with the real development of these subjects. 
167. Under the head of conceptions to be defined we 


have hitherto considered only comparatively simple ones, 
conceptions of figures, things, properties, and easily in- 
telligible relations : but among the words used in speech, 
every one of which may under certain circumstances call 
for a definition, we often find very complex relations 
between a great variety of points of attachment com- 
prehended in one simple expression. No one who was 
not hide-bound by prejudice would require that the ex- 
planation of such conceptions should take the regular form 
of a simple definition ; and to find special names for all 
the other very various methods which may be employed 
would be nothing but useless pedantry. The universal 
principle of applied logic is simply that all ways are 
allowable which lead to the goal ; it hopes for no more 
than to remove our doubts as to which way is passable 
right up to the end, and which not, by pointing out that 
w.hich has long ago been tested : it never forbids our 
seeking new ways to satisfy new needs. It is always 
allowable therefore to begin with a preliminary description, 
with comparisons and analogies, with discussions of any 
kind, in order to familiarise the hearers with the meaning 
of the subsidiary ideas we wish to employ and the peculiar 
combinations we wish to establish among them, and having 
thus prepared the way to proceed to set forth what we 
wished tp explain in a formula which is brief and intelligible, 
though it presupposes what has gone before and cannot be 
separated from it. 

This reminds us however of another twofold division of 
all definitions. We may characterise M by the aggregate 
of marks displayed by the conception when it is present to 
our minds in its completeness : this kind of definition, 
which we illustrated just now in the case of the circle, may 
be called descriptive definition : we have recourse to it 
mainly in the case of actual things which we only know 
from the outside and whose definition therefore is in fact 
nothing but a methodical description. But we can also fix 



M by pointing out a way in which, riot by the mere addition 
of other ideas, but by freely using and manipulating them 
at will, this idea can be produced with certainty. This I 
would call genetic definition, understanding thereby (and 
this I wish particularly to emphasise) not a statement of 
the process by which the content of the conception M is 
actually found, but only an indication of the way in which 
the mental picture of this content M may or must be formed. 
' Let a straight line revolve in one plane about one of its 
extremities, and combine the successive positions of the 
other extremity:' that is a genetic definition of a circle. 
The circle as such is not made at all : but supposing a 
particular circle such as we draw to have been already 
made in some way or other, we may certainly form a 
mental picture of it in the way indicated by this definition. 
But we may form that mental picture equally well by sup- 
posing the length of the two axes of an ellipse to alter till 
both are equal to r ; or by supposing a cone to be inter- 
sected by a plane at right angles to its axis. And thus an 
idea, whose content has in itself no genesis, may admit not 
only of one, but of so many genetic definitions as there are 
ways of forming the idea of this content by the manipula- 
tion of other ideas. Among these genetic definitions then, 
using the term in a somewhat extended sense, we may 
include the above-mentioned miscellaneous methods : they 
try by indirect means to make us form a mental picture of 
M) when it is impossible or inconvenient to say directly 
what M is. 

168. Strictly speaking, whenever we undertake to define 
a conception J/, our aim is to give it a higher degree of 
definitcncss than it has yet. But in fact the problem 
usually narrows itself to the transformation of a clear idea 
(clara perceptio) which we already have of J/, into a distinct 
one (distincta\ or of a mere mental picture, which does but 
comprehend M in a loose general way as a connected whole 
made up of parts which are familiar, into a real conception of 


M. These two expressions may be regarded as equivalent. 
For according to old established usage we are justified 
in saying we have a clear idea of anything when we think of 
it as one, and as a connected whole, and lastly as dis- 
tinguished from others with precision enough to avoid 
confusion; but it does not become distinct till to this 
is added the general law which regulates the connexion 
of the parts, and further the characteristics which it has 
in common with other species of a certain genus, and lastly 
those particular characteristics which distinguish it from all 
the other species of its own genus. In treating of Pure 
Logic we identified this increase in definiteness with the 
transition (in technical language) from an idea or mental 
picture to the conception or actual comprehension of a 

But now there are cases in which our idea of an M which 
is to be defined is far from possessing the clearness here 
supposed : names are handed down to us which have 
become part of our language though their meaning has 
never been precisely fixed. Thus we speak of virtue and 
sin, of good and the highest good, of appearance and 
reality, with a full conviction that we mean something 
very definite by these names, and ready to draw important 
inferences from them in reference to that to which we 
apply them. But at last the difficulties in which we en- 
tangle ourselves convince us that strictly speaking we did 
not know precisely what we meant, that we had not com- 
pletely fixed the conditions which must be satisfied in order 
to justify the application of these names, that we had in 
short trusted to hazy ideas, the clearing up of which 
is of the very first importance. This we try to effect 
in a very simple way. If we were entirely ignorant of the 
meaning which M was intended to* bear, we should have 
no means of finding it out ; but also it would never have 
occurred to us to apply this name had not some part of 
its meaning (say a) been fixed beyond a doubt that very 



part namely which now impels us to use the term the rest of 
whose meaning is still hazy. This a we first take tentatively 
as a complete definition of J/", and consider whether a cor- 
responds to what we mean by M. It is a matter of common 
experience that in cases where we are not in a position 
to express the meaning of M in positive terms we may yet 
see whether an idea a that is offered as a definition of 
it is adequate or not. Thus when we are trying in vain 
to recollect a name we can yet pronounce with perfect 
certainty that a suggested name is not the right one ; and 
further any resemblance it may have to the right one 
makes an impression on us, and sometimes reminds us 
at once of what we want, at any rate it helps to make 
plain the other points in which the right name differs from 
the suggested one. We are in the same case here : a is not 
utterly wrong and incapable of comparison with M\ the 
comparison of the two therefore does not lead to the 
bare negation of their identity, but puts us on the track 
of a supplementary b which must be added to <?, or an 
alteration b which must be effected in a in order to make it 
answer exactly to M. Now putting M down as equal 
to a + b we make a second attempt and repeat the same 
course of comparing and supplementing by fresh terms 
c and </, till at last we get a definition M =. a + b + c + d 
which in its expanded sum of characteristics exactly co- 
incides with what we meant by M. In this very simple 
process of thought rather than in a strictly inductive method 
lay the art which the Platonic Socrates used ages ago to 
clear up hazy conceptions. 


Of the limitation of Conceptions. 

169. IN the course of an investigation we may be led 
by a definite purpose to trace a group of characteristics 
i k I through all the otherwise different objects in which 
it occurs, and to ask what influence is exercised by its 
presence upon the rest of their characteristics. The result 
of this comparison then will itself teach us whether the 
other characteristics which each of these subjects has in 
virtue of the genus to which it belongs are modified by the 
presence of / k I in any remarkable and particularly in any 
constant manner. If this is the case we often form 
out of / k I and out of the idea of a more or less pre- 
cisely determined subject a new generic conception M^ 
treating all the ideas in which i k I occurs as species of 
M. But 'whenever this is not the case (and not seldom 
too when it is) we content ourselves with treating the 
presence of //as one of the countless variable conditions, 
which affect other ideas so far as to necessitate certain 
alterations in them, but do not themselves form a generic 
conception under which the several instances in which 
they occur could be arranged as species. Now a living 
language is believed by those who use it to have already 
sufficiently distinguished in the coinage of its words the two 
kinds of cases in which these two methods are severally 
appropriate. Of course they will allow that enquiry, as 
it goes deeper and deeper, will discover many a new group 



of characteristics ikl having such a decisive influence upon 
the whole bearings of every conception that contains it as to 
make it worth while to erect this group into a separate 
generic conception M and to mark it by a name : and 
language is in fact constantly enriching itself by new names 
for the ideas thus newly discovered. But, on the other 
hand, they will also assert that none of the conceptions 
already found and fixed by the creation of a name are 
unworthy of this distinction : each, they insist, really means 
something coherent, which is thus justly cut off, as a whole 
with well marked boundaries, from all other similarly co- 
herent ideas. 

17O. These conceptions which our inherited language 
supplies are the tools with which our thought must work 
and that not merely because we have no means of com- 
munication except the words which have been invented 
to express them : in this store of words is treasured up 
the concentrated result of the thought which the human 
mind has from the earliest times bestowed upon the world 
to which it has access, and we may suppose that the same 
impulses which led it to fix its conceptions in this form 
would also in the first instance assert themselves in us were 
we to go through the same labour. 

But that these impulses, however natural they may be to 
man, yet leave room for doubt is shown by the divergence 
that constantly occurs in the application of the conceptions 
thus formed. When the question arises whether some 
predicate P is to be affirmed or denied of a subject S, one 
maintains that is a kind of M and therefore is a P; 
another objects that S is no M and therefore no P ; a third 
allows that S is indeed no M 9 but an JV, but declares that 
this does not matter, and that what is true of M holds 
good of N also, while a fourth insists that the difference 
between M&nd N establishes a difference between the two 
in respect of P. 

The divergence that here shows itself culminates in two 


opposite tendencies, dominating the whole of our thought. 
The one is a tendency to exaggerate every difference that 
presents itself into absolute difference, and with the familiar 
formula ' this is something quite different ' to resist all ar- 
gument from one case a to another case b which resembles 
a but is not exactly like it : this tendency becomes in life 
and in science the spirit of the pedant and the philistine. 
The other is a tendency to ignore the fact that a difference 
which is not absolute difference may yet have a qualified 
value, and with the barren phrase ' all is one at bottom ' 
to obliterate all the fixed boundaries which define the 
province of each conception, thereby destroying the only 
grounds upon which certain predicates are attached to 
certain subjects and to no others : this becomes in thought 
and action the principle of a no less ruinous libertinism. 
A glance at the momentous consequences of these con- 
fusions makes us alive to the necessity of clearly under- 
standing what reasons there are to justify us in dividing 
the whole extent of the intelligible world into definite 
conceptions, where the boundaries of their several provinces 
are to be drawn, and what value is to be assigned to this 

171. We are led to very various issues by the attempt 
to answer these questions even where they are easiest and 
least pre n sing, viz. in regard to the simple contents of 
sensuous impressions. We have a right to assume absolute 
difference between simple sensations A B Cwhen we cannot 
imagine any intermediate steps by which the peculiarity 
of one could gradually pass over into that of another, 
and when further we cannot think of any mixture of two of 
them which would give a new simple sensation, and when 
lastly there are no degrees of contrast between them such as 
would enable us to estimate the difference between A and 
B as greater or less than that between A and C or between 
B and C. We find these relations, or rather this lack of 
any assignable relation, between A B and C if A stand for 

Q 2 



colour, B for sound, and C for smell. We may keep the 
old name and call them disparate or incomparable. 

This conclusion will not be affected by various secondary 
considerations which may be urged. It may be pointed 
out, for instance, that all three exist only as states of our 
consciousness. To this we reply that they all are indeed 
sensations, and may be called, according to the usage of 
logic, species of sensation ; but that the conception of sen- 
sation in general cannot serve here as a generic conception 
in the sense of supplying a law for formation. When we 
think of the shape of an obtuse-angled triangle as subordi- 
nated to the general conception of a triangle, we have in 
the latter a constructive formula, whose application has but 
to be varied within its own limits in order to show us that 
there are right-angled and acute triangles besides that one 
species from which we started. But the subsumption of 
colour under the general idea of sensation (for it is only 
subsumption that is possible here, not subordination) can 
never enable us to conclude from this general idea that 
there are such sensations as sounds and smells besides 
colours. Although these three then are, to use the ordi- 
nary phrase, kinds of sensation, yet within the limits of 
this universal they remain quite disparate the one from the 

Again as states, as motions or affections of the $oul, these 
various kinds of sensation may produce certain secondary 
effects that are comparable with one another, and it is 
certainly allowable on that account to compare a certain 
colour a 1 with a certain sound b* or a certain smell c l : but 
still that which produces these comparable after-effects 
remains itself quite incomparable. And we must make the 
same reply to the physicist and the physiologist, when the 
processes which must take place in the outer world or in 
our nerves in order to produce the various kinds of sensa- 
tion are traced back by them to comparable, or perhaps 
even to closely allied movements of material particles : 


they must conclude not with the curious assertion that 
there is therefore strictly speaking no qualitative difference 
between these sensations, but rather with this other asser- 
tion which is true, viz. that in spite of the similarity of 
origin there is not the slightest similarity in the results. 
There is no room for doubt here, except in so far as the 
unprejudiced observation of ourselves, which is here the 
sole criterion, is unable to pronounce decidedly. This is 
the case with regard to taste and smell. Sourness is un- 
doubtedly common to both; but the other sensations of 
taste and smell also seem to form a connected group, only 
that some members of this group are excited only by the 
agency of liquids, others only by that of gaseous matter. 
It may be that the sensations of these two senses, which on 
this account must have different organs, are themselves 
homogeneous and distinguished only by secondary sensa- 
tions dependent upon the position, shape and action of 
their respective organs. But it is not the business of logic 
to decide this question : all we need do here is to warn the 
reader when he has a direct perception that two modes of 
consciousness are incomparable, never to allow this to be 
overborne by sophistic arguments based upon the similarity 
of their antecedents or consequents. 

172. The other question, not as to our right to separate 
A and ^,,-but as to our right to join together all that we 
comprehend under A, calls for a similar remark. For a 
long time people tried to dazzle the public with the stupid 
paradox that black and white were no colours because they 
did not like the prismatic colours depend upon a definite 
number of undulations of light. The progress made of late 
in the physiology of vision has completely cut away this 
ground ; but even if this had not been done, no one could 
have had the right to override language in this fashion. 
Long before we knew anything about the exciting causes of 
our sensations, language had invented the name of colour 
for a group of sensations which by a homogeneous quality 



directly perceived and undeniable, viz. by shining or what- 
ever else we like to call it, are at once bound together and 
separated from tones that ring or resound and scents that 
are smelt Granted that the name shining is only appro- 
priate to white and not to black, still the fact that the 
fundamental quality thus imperfectly designated is shared 
by both in common with the other colours admits only of a 
verbal not of a real denial, and the common usage of the 
term colour so as to include both was therefore completely 
justified against the unsupported objections of the savants. 
In other fields also we find similar instances of the en- 
croachments of scientific theory, not always harmless in 
their results. Thus chemistry for a long time contributed 
to the confusion of speech by identifying oxidation and 
burning. Men assuredly spoke of burning long before they 
knew of oxygen, and always meant by it a process accom- 
panied by visible light and sensible heat, which permanently 
altered the constitution of a body : a glowing iron rod there- 
fore was not said to burn, because no lasting alteration was 
found in it when cooled : but also such a permanent change 
would not have entitled the process which produced it to 
the name of burning, in the absence of the sensible develop- 
ment of flame and heat. The notion of burning then by no 
means coincides with that of oxidation : many substances 
are oxidized without burning, and on the other hand, when 
heated antimony is immersed in gaseous chlorine and com- 
bines with chlorine, throwing out flames the while, this 
process is undoubtedly one of burning though not oxidation. 
Geometers, again, knew ages ago that any system conceived 
in abstract terms, i.e. arithmetically, provided that not more 
than three scales be required for the arrangement of its 
various elements, may be presented to our perceptions by 
means of spatial constructions. Now there is nothing to 
prevent a mathematician from c'onceiving systems based 
upon any number of scales greater than three, only it is 
plain that such systems can no longer be envisaged in space, 


and that the name * dimensions ' which could be applied to 
the scales in its ordinary sense of dimensions of space so 
long as they were only three, can now bear only the more 
abstract sense which I tried to express by calling them 
scales. As space therefore means for us nothing but a 
system that we envisage in this peculiar way which certainly 
cannot be derived from any considerations of mere number, 
to continue to speak of a system of four or five dimensions 
as space is but to make sport of logical distinctions. Let 
us be on our guard against all such attempts : they are 
nothing but scientific freaks, which intimidate the popular 
consciousness by utterly useless paradoxes and make it 
doubt its well established rights in drawing the boundaries 
of its conceptions. 

173. When we now ask how the several coherent members 
of one of those disparate kinds of content A B and C are 
related to one another, we find that these relations are 
peculiar and not always of the same kind. No one has yet 
succeeded in reducing the several kinds of taste to a satis- 
factory system : but the path which common usage takes in 
naming them, incomplete though that nomenclature be, 
seems to me the right path. Certain primary forms are 
distinguished by names of their own, as sweet //, sour y, 
bitter TT, and the others such as sour-sweet v ^ bitter-sweet 
/x TT, are regarded as compounds of those well-marked 
primary tastes. Our imagination could never have lit upon 
this mode of naming them had it not been guided thereto 
by the direct impressions of sense, for we cannot make 
differences unless they are already present actually or 
potentially in the data. Now these names imply that they 
are actually present, not of course in the sense that the 
sour-sweet is an aggregate of a sour and a sweet that can be 
separated as much as if they were tasted at different times, 
but in the sense in which we speak of a mixture as opposed 
to an aggregate. The fact that such a mixture is possible 
here, i.e. that sour and sweet may be united in one im- 


pression in a manner that we can scarcely describe but 
easily feel, while sweet and red cannot, distinguishes the 
relation of the several tastes to one another from that of the 
disparate groups ABC. 

It may be objected that in the sour-sweet the difference 
between the sour and the sweet is only present potentially 
not actually; that there may easily be a third impression o>, 
itself simple and in no way compound, yet forming a con- 
necting link between /x and v, and that this then, on account 
of its resemblance to both, is designated in speech by the 
two limits /x and v between which it falls, without implying 
that it actually is a mixture of the two. This objection 
I should not consider sound unless there were present in o> 
besides that in which it resembles /x and v an independent 
remainder that could not be accounted for by the combina- 
tion of /x and v ; where this is not the case this third im- 
pression o> will not merely be called a mixture /x v by an 
arbitrary freak of fancy, but will in fact be that and nothing 
else. But the primary forms /x v TT and all mixtures of these, 
though made one group by the fact that they all alike 
appeal to the sense of taste C, yet within those limits can 
only be regarded as disparate from one another. A man 
who had tasted nothing but sweet could never by any con- 
ceivable modification of the feeling it gave him discover 
the peculiar nature of sour or bitter that he had not yet 
experienced. There is then no transition from /x "to v or TT 
through independent connecting links, but we must first 
know fx v and TT and then get the intermediate links by 
various mixtures of these. 

We find the same relations between colours, and I took 
occasion in an earlier passage 1 to justify the common usage 
of speech in always distinguishing a limited number of 
primary colours, and inserting the rest as mixed colours 
between them. It is of course possible to lead the eye 
gradually through skilfully selected middle-tints from the 
1 [Sect. 13.] 


impression of one colour to that of another : but while red 
passes into orange or violet only by an admixture of yellow 
or blue which can still be felt as yellow or blue, that which 
makes red what it is does not pass over into that which 
makes blue what it is. A man who had experienced one but 
not the other could never discover in the simple nature of 
red anything which could possibly be modified, heightened, 
or cooled down in such a way as to lead him to imagine 
what blue is : he would have to learn what blue is before he 
could mix the two extremes together so as to arrive at the 
intermediate violet. The modifications of which the several 
primary colours are capable must also be regarded in the 
same way. We undoubtedly have the right to consider 
bright blue and dark blue as kinds of the same bJue : but 
these kinds also are produced by the mixture of white or 
black with a pure blue that is always the same though never 
visible in its purity. Only I would once more briefly re- 
mind the reader that all that I have hitherto said refers only 
to the nature of our sensations after they have arisen in our 
consciousness, and has nothing to do with the physical or 
psychical conditions of the act of sensation. 

174. With sounds the case is essentially different. After 
a comparison of several sounds we distinguish first of all 
three predicates. The peculiar tone of the instrument 
which is sounding, whatever the physical antecedents may 
be, is for our feeling a simple property which defies further 
analysis, more analogous to a taste than to anything else. 
However strongly we may be moved by the secondary 
effects of this peculiar tone the essential nature of the note 
seems to us to be quite independent of this, and also of its 
second property, viz. its loudness or strength : we regard 
both only as ways of producing the same note, the dis- 
tinctive nature of which lies in its pitch. But in this third 
aspect sounds do not like colours fall into a number of 
distinct stages, such that one can pass into another only by 
mixture, but they rather form a continuous series, in which 


the difference between two more distant members is only 
a multiplication of the difference between two adjacent 
members. It is impossible to make a proportion in which 
red shall stand to blue as yellow to any fourth colour : but 
the difference between two notes can always be stated as a 
multiple of some difference which we take as the unit. 
This difference itself is of a quite peculiar kind : we should 
not use the phrase * higher ' and ' lower ' in speaking of 
sounds, unless, quite apart from the frequency of the sound- 
waves which we certainly do not feel, our feelings themselves 
announced one note as a heightening of another : but this 
quantitative idea cannot be referred here as it can elsewhere 
to a qualitative content that is independent of it : a note d 
is different from another c even in quality just because in 
it the undefmable common property of sounding which it 
hhares with c is c heightened ' in that peculiar way which we 
can only express by this happy metaphor, or at most by the 
more technical phrase * qualitative intensity.' The differ- 
ences of notes therefore are homogeneous and measurable 
in extent, which the differences of colours were not : the 
notes intermediate between two others arc not formed by 
mixing these two together, but are on a footing of perfect 
equality, as original members of the series, with those 
members between which we place them. 

And lastly the whole scries is endless : it is not possible, 
in addition to the colours known by experience, td imagine 
a new colour of which we can have an idea though it 
happens that our eyes never saw it ; the scale of sounds, on 
the contrary, may be continued ad infinitum because each 
is generated out of its predecessor by a heightening which 
is felt to be homogeneous. It is not unmeaning to talk 
of sounds higher or lower than any that can ever come 
within our experience, because we have here (what we could 
not have if we tried to imagine new colours) a distinct idea 
of the way in which these sounds would differentiate them- 
selves 2/"they were audible. 


175. With some modifications, which I leave the reader 
to make, these remarks apply also to the series of our 
sensations of heat : but at the same time the latter exhibit 
a new feature. The living body's own need for warmth 
gives a peculiar significance to certain sections of the 
series ; we distinguish cold, cool, lukewarm, warm, hot, and 
fancy that these terms have a definite meaning; but not 
only would it be impossible to draw a hard and fast line 
where cool ends for everybody and lukewarm begins, but 
even if we interrogate our own feelings merely we are 
obliged to confess that there must be a certain caprice in 
choosing the one name or the other. We may connect with 
this contrast of heat and cold, and of high and low sounds, 
a great number of other pairs of ideas, the content of which 
is not so directly derived from sensations, e.g. great and 
small, strong and weak, many and few, old and young, and 
many more of the same sort. 

However decided a contrast is intended by the two terms 
of these antitheses it is always impossible to mark off the 
province of the one from the other, they constantly and 
insensibly pass into one another. But when we go through 
such a scries the passage from a to z and that from z to a 
are very clearly different, -to some extent they admit of 
definition, and our immediate feeling at any rate never fails 
to distinguish them. We cannot say what is warm nor 
what is cold, but we can say without any doubt whether a 
is warmer or colder than b : in this case the decision is 
a matter of sensation ; in passing from a to b we are con- 
scious of a change which is the opposite of that which we 
experience in passing from b to a. We cannot say what 
great and small mean, but the statement that a is greater 
than b is quite free from ambiguity, and may be defined to 
mean that if b is taken from a there is left a positive re- 
mainder <5. And it is the same with the other examples : 
these adjectives are all derived not from the apprehension 
of one idea but from the comparison of several, and denote 


relations which have no fixed value or meaning apart from 
a second point of comparison. These adjectives therefore 
are indefinite in the positive ; only their comparatives have 
an unambiguous meaning. Where the positive form is used 
in speech it means that the comparative term may be applied 
to the thing denoted when compared with an unexpressed 
standard, which either in the estimation of the speaker or 
in common opinion is the normal or usual state of the thing 
in question. 

176. There is one more point to consider in connexion 
with sound and sensations of heat. Sounds being in them- 
selves of perfectly equal value we have no inducement to 
select some few of them as fixed points and to give them 
prominence by naming them. But on aesthetic grounds 
we want to articulate the whole series. As the simple sen- 
sation of a note is undefmable we characterise it by stating 
the cause which will at any moment produce precisely that 
note, i.e. the frequency of the vibrations upon which it de- 
pends. But there is no reason for preferring one number 
to another, and as every member of the series may be 
defined with equal ease in the way named, the musical scale 
has in fact no absolute starting-point. It is true that other 
circuinstances, viz. the harmonic relations of notes, which 
I must here pass over in spite of the interest which they 
have even for the logician, lead us to arrange the series 
in octaves ; but even this arrangement has no fixed starting- 
point ; we may begin at any height we please. 

Our sensations of heat do not admit of such a simple 
definition by their causes ; we are obliged to have recourse 
to the other observable effects of their unknown cause, viz. 
the expansion and contraction of bodies. To take the 
melting-point of ice as the point from which the degrees of 
temperature should be measured in an ascending and 
descending scale was to choose a quite arbitrary zero to 
reckon from, though one very well adapted to its purpose : 
for the fluidity or solidity of water is a point of cardinal 


importance in the meteoric and organic processes which 
surround us. But it is after all merely a zero in our cal- 
culation, not in the thing calculated. Starting from the 
unknown amount of heat (call it x) which is present at the 
melting-point of ice, all we do is to reckon the increase or 
diminution of this amount by multiples of a unit-degree 
chosen expressly for this purpose. Thus 12 is not the 
double of 6, but the difference between o (which is equal 
to x) and 12 (which is equal to x -f 12 units) is twice as 
great as the difference between o or x and 6 (which is 
equal to x -f 6 units). 

The reader may see by this simple illustration that though 
a series or a complex system cannot be articulated and ar- 
ranged in a regular order unless there be a corresponding 
regularity in its own relations, yet thought frequently has to 
take a quite arbitrary starting-point and an arbitrary stan- 
dard in order to master and make use of this regularity; 
and that such an arbitrary arrangement, though admitted by 
the nature of the object and justified in its results, yet must 
not be looked upon as a property inherent in the object 

177. Practical life offers many illustrations of this remark. 
We here have to do with qualities which either attach to 
various persons and things in very varying degrees, or which 
in one an.d the same subject take successively a continuous 
series of values, from which proportionate effects are ex- 
pected. But it is only in nature that effects vary con- 
tinuously in accordance with the conditions : where the 
result does not follow till it is produced by human action, 
the exact observance of the desired proportion is generally 
prevented by the fact that the labour required would be out 
of all relation to the end in view. We have to content 
ourselves with breaking up the whole series of values into 
sections and acting as if the conditions were the same 
throughout each section, fixing the result at an average 
amount, which will be too great for the first and too small 



for the last members of that section of the series. Thus for 
the purposes of taxation we divide the series of properties, 
from absolute poverty up to the highest pitch of wealth that 
is likely to be found, into a number of classes; in calculating 
the premium to be paid on a life-insurance we reckon age 
by years or at lowest by some considerable fraction of a 
year ; in calculating interest we keep to a day as an in- 
divisible unit. Again it may happen that a quality gradually 
attains a certain pitch to the attainment of which we desire 
to attach certain consequences, though we cannot say at 
what moment the decisive condition is fulfilled. That 
maturity of body and mind which we have in our minds 
when we say that a man is of full age or has attained his 
majority is certainly attained by different persons at different 
times of life ; but it is impossible to find out the actual 
moment in each individual case, not merely because it 
would necessarily be an endless business to appraise the 
total merit of the person, nor yet because such a censorious 
proceeding would be unjustifiable, but because, though the 
higher grades of maturity and immaturity are easily recog- 
nisable, there is really no certain mark to distinguish them 
in doubtful cases. But for all that the needs of social life 
require that a definite time be fixed ; so the law has to fix 
it summarily, and attaches to the completion of certain days 
and hours the beginning of certain rights and duties, though 
no one supposes that the capacity and the obligation which 
were absent yesterday have actually sprung up in the course 
of the night. But though this proceeding is summary it is 
not without reason : the choice is limited to times which 
correspond without any appreciable difference in accuracy 
to the requirements of the situation ; all that is arbitrary is 
the preference of one out of a number that would all do 
equally well. 

There are other cases in which we are still further from 
finding any precise standard in the nature of that which has 
to be settled, and must look for it in the further ends 


whose attainment is to be facilitated by the settlement. 
Such are the fixed periods within which certain conditions 
must be satisfied in order to establish some legal claim or 
to avoid some legal obligation : though the outlines of these 
arrangements are determined by the object specified, their 
details aim at nothing but logical precision. This our 
ancestors effected by not measuring the more important 
periods by entire units of time of the larger kind, but 
adding to such units some fraction of them, some days to 
a week, some hours to a day; by these means they nar- 
rowed the period within which (to use a common but 
rather loose phrase) a man might have fancied that he 
was satisfying the law. The police again are quite right 
when in order to prevent disturbance of the peace they 
summarily fix the number of persons that shall be held 
to constitute a forbidden assembly at three or five, thereby 
barring disputes like those the old sophists used to raise 
when they asked how many grains of corn are required 
to make a heap, or how many hairs must be lost to make 
a bald-head. 

178. To return from this digression : -whether a note is 
to be called high or low, a liquid hot or cold, are questions 
that people never quarrel about : there are no interests 
attaching to the content of these conceptions that could 
make us -hesitate to admit at once that their meaning is, as 
we said, relative. It is different with good and bad. We 
set the highest value on the fixity and absoluteness of 
these conceptions : every action, not simply as compared 
with others but as it is in itself, must it is thought be 
unequivocally included in the one and excluded from the 
other ; people even think they are bound to deny that 
there are any degrees of goodness in the good or of 
badness in the bad, for fear lest the diminishing values of 
the two should at last meet in the indifferent as a zero- 
point, and a constant transition be thus set up between 
two opposites which ought rather to be severed by breaking 



down every bridge. But this logical rigour is utterly at 
variance with the unprejudiced judgment which we all bow 
to in real life. No one really doubts that there are degrees 
of goodness and badness, and no one can persuade us that 
no acts are indifferent till he has artificially limited the 
conception of an act. But it really is no use to try to fend 
off the threatened confusion of good and bad by first 
dividing all actions into those which can be judged morally 
and those which cannot, and then proceeding confidently 
to divide the former into two absolutely opposed groups, 
the good and the bad. We thereby do but move our 
doubts a step further back ; for the question now is where 
is the line to be drawn between that which calls for a 
moral judgment and that which does not ; and this line 
as before will seem to vanish in a perpetual passing of the 
one into the other. 

Again the relation of the pleasant to the beautiful and 
the good, though a less pressing question, is one of great 
interest on aesthetic grounds. To the man without a 
theory they seem to arrange themselves in an ascending 
series, not merely according to their value but according 
to the meaning of their content ; not of course in the sense 
that by mere intensification what is extremely pleasant 
would become beautiful, or the highest beauty pass into 
the lowest grade of goodness, but in the sense that there 
are kinds of the pleasant, distinct in quality, which begin to 
have a right to the name of beautiful, and forms of beauty 
which produce an aesthetic impression akin to moral ap- 
probation. But those who theorise upon morality and 
upon art alike resist this admission ; they deem the beauti- 
ful falsified if it has anything to do with the good, the good 
degraded if it has anything in common with the beautiful, 
and through this with the pleasant. Here too, with regard 
to beauty at least, people have been found to deny all 
differences of degree, and to maintain that what is beautiful 
at all is entirely beautiful, and that if you allow there is 

Chap. II.] TRANSITIONS. 241 

anything more beautiful, you cannot think this really beau- 
tiful at all. 

179. Let us, in order to settle these doubts, look around 
for other illustrations. Of the straight line, from its nature, 
there is of course but one species known to the geometer ; 
but in curves he distinguishes countless degrees of curva- 
ture of measurable value, so much so that the straight line 
itself appears as the extreme limit to which the curve con- 
stantly approximates as the radius increases. Yet in spite 
of this unbroken continuity not merely does the geometer 
persist in the general statement that curved and straight 
are opposites that can never be reconciled, but no doubt 
ever arises in its application to a particular line which is 
accurately known ; however near it may come to a straight 
line it is yet quite undeniably curved, so long as the radius 
of curvature has any finite magnitude. 

Again a curve may in one portion of its course be con- 
cave to an axis to which it is convex in a further portion ; 
if it makes this change of direction in an uninterrupted 
sweep without any angle that breaks the continuity, there 
is no doubt that its tangent at the turning-point, and there- 
fore the element of the line itself, is parallel to the axis in 
question, and so neither concave nor convex ; but although 
both directions thus visibly meet in one zero point of 
indifference that belongs to neither, yet the opposition 
between them is thereby neither altered nor removed ; on 
this side of that point the curve remains entirely concave, 
on that side entirely convex. Take a simple instance : 
between i and 2 we may insert countless fractions rising 
gradually in value from i to 2 ; between full daylight and 
midnight darkness countless degrees of illumination not 
only are conceivable but actually occur ; between pleasure 
and pain there lies an uninterrupted series of feelings which 
connect the one with the other : but i does not on that 
account become equal to 2, nor do darkness and pain 
cease to form a perfect contrast to light and pleasure ; and 



at the same time each member of these pairs, by itself and 
without reference to the other member, is something so 
definite that we never mistake the one for the other. These 
illustrations are sufficient to explain the statement that the 
existence of countless degrees through which two opposites 
A and B pass till they meet in a common zero-point of 
indifference, does not destroy the difference or opposition 
between the meanings of A and B themselves. 

180. And so even if the moral philosophers had succeeded 
(and it is their business and not ours at present) in deter- 
mining what they mean by good A and bad B as precisely 
as the geometer defines what he means by convex and 
concave, they would still have had no ground for denying 
that good and bad have degrees and meet in the indifferent, 
in order to maintain unimpaired the distinction between 
the two. The specific meanings of the general conceptions 
good and bad are not in the least degree altered because 
particular cases to which the terms are applied partake 
more or less fully of the character of one or the other of 
these opposites. But the zero-point of indifference can still 
less contribute to the confusion of the two, for its meaning 
is not that both are true at this point, but that neither is 
true ; it is therefore merely a point of separation, on this 
side is only good, on that side only bad. 

On the one hand then the maintenance of the distinction 
between good and bad is no reason why people should 
deny that there are degrees of good and bad ; on the other 
hand we must insist upon an explicit admission of the fact 
that there are degrees. To deny it, to repeat the old Stoic 
paradox omnia peccata esse aequalia, or to go on preaching 
that even the smallest error is still not truth but error and 
nothing else, is but to waste time in tedious assertions which 
as they contain only half-truths may on this very principle 
be called errors and nothing else. It is not true that ,a 
curve is once for all a curve, so that the degree of its con- 
vexity or concavity is quite a secondary consideration, 


which has nothing to do with its character as a curve ; the 
fact is that one curved line is actually more curved than 
another, and so realises more intensely the character 
common to both. Similarly the good or bad intention 
out of which an action springs can not only be measured 
in a secondary way by the importance of the interests 
affected by the act or of the circumstances under which it 
is done, but can itself be estimated according to its degree 
of goodness or badness; for such an intention is by no 
means a mere form which is alike in all cases ; it is an 
inner process which not only must reach a certain degree 
of intensity in order to generate the impulse which every 
act requires or to overcome certain obstacles, but has also 
a certain degree of value according to the amount of the 
good or evil which it consciously aims at producing. Error 
again is not merely not-truth ; that would not distinguish it 
from doubt ; it is a departure from truth, and has therefore 
a measurable magnitude, indeed is inconceivable without 
it ; a man whose thoughts are occupied with real problems 
therefore will not be so silly as to reject in identical terms, 
as mere errors, two assumptions of which the one is so far 
from the truth that it leads to no knowledge at all, and the 
other so near that it leads to nearly all the knowledge of 
the subject that can be expected. 

181. It may be that the series of the pleasant, the beauti- 
ful, and the good (the further consideration of which I 
leave to the reader) has already suggested another relation 
that can exist between a series of conceptions, which I will 
first of all illustrate from geometry. Imagine two pyramids 
A and B presenting similar horizontal sections but one 
sloping more steeply than the other ; if we place them so 
that the apex of the one (the less steep) lies within the 
other (the steeper) and upon. 1 a point in its axis, then the 
plane which passes through the intersection of their surfaces 
belongs both to the series of planes of which A is the 
integral and to the series of other planes the endless succes- 

R 2 


sion of which is summed up in B : similarly we can imagine 
a third pyramid C which should in like manner have a 
plane in common with B. 

Now the generating law of each of these solids, with 
reference to the common axis of all three and the position 
of the apex in that axis, may be stated in a formula, which 
would have to be compared with the general conception of 
A B and C respectively. It would then appear that in the 
A series there is one member that also satisfies the require- 
ments of B \ and therefore that as to this member it is a 
matter of doubt or of indifference whether it is to be classed 
under the conception A or .Z?, not because it satisfies 
neither, but because it perfectly satisfies both at once. But 
with the exception of this particular case all the instances 
of A, all the other planes by which the compound solid thus 
formed could be intersected, would belong exclusively 
either to A or to B. The same would be true of the plane 
common to B and C. 

In these cases then it is due to the very nature of the 
essentially distinct conceptions that certain members of the 
series which they severally characterise become ambiguous, 
so that by themselves and without taking count of some 
secondary point, such as the manner of their origin and 
development, it is not safe to ascribe them exclusively to 
any one of these conceptions, though apart frm these 
particular cases there is no doubt at all about the difference 
of the conceptions. We have here named ABC and so 
expressed them as conceptions, leaving the particular cases 
unnamed. But the purposes of speech may sometimes 
suggest the opposite procedure. We may name and fix 
certain conceptions M N O which have quite unambiguous 
and distinct meanings only in particular cases, which we 
may picture to ourselves as salient points, as maxima or 
minima, in a connected series. We shall then find the 
reverse of what we found just now, i. e. we shall find many 
contents furnished by feeling and experience which have a 



place indeed between two of these conceptions, but only 
between them, corresponding completely to neither. 

182. As illustrations of the latter procedure we may take 
compound conceptions got by starting not from one but 
from many points of comparison at once. With such a 
conception no doubt every instance agrees which in each of 
these respects is found to have the appropriate mark ; but 
the applicability of the conception becomes doubtful in 
many other cases, which from one point of view would 
certainly be included under it, but from another which 
must also be considered would certainly not. Various 
thoughts thus cross one another in the conception of illness. 
Illness is certainly above all things a departure of the bodily 
condition from a supposed fixed standard. But a mal- 
formation, which departs considerably from the natural 
structure of the body, still cannot be called an illness, so 
long as it does not impair the vital functions, nor so long 
as it remains constant and runs no natural course through 
various stages. A wound always in some degree alters 
structure and function, and also runs a natural course ; but 
a slight wound is not called an illness, plainly because it 
does not involve danger nor make the body unserviceable 
for any important purposes of life ; but again a very severe 
wound is also not called an illness though it does both ; its 
origin is, too sudden and too entirely due to external 
violence, and now we observe that when we spoke of 
illness, we thought of a state which, though dependent 
upon some external cause for its origin, yet takes its definite 
shape from the peculiar interaction of the internal forces. 
But now a cold is such a reaction of the internal forces against 
an external stimulus : but a cold is scarcely called an illness 
so long as the element of danger is absent : and just as we 
here help ourselves out with the milder phrase f unwell/ so 
we use the term health with a certain latitude, allowing 
room for the slow advance of a number of disturbances 
connected with individual idiosyncrasies. 


It is not difficult to say what is the right course here. It 
is impossible in such cases to find a definition which shall 
be in harmony at once with the requirements of science 
and with these strange caprices of language : if we want to 
determine the conception, we must disregard usage and fix 
it arbitrarily. In the instance we have chosen this is* scarcely 
needed, for pathology gets on very well without any unim- 
peachable definition of the nature of illness in general ; and 
the physician has absolutely no need for logical generalities 
which yield no guidance in practice. 

But in other cases it is not so. In our conception of 
crime all sorts of consideration cross one another, we con- 
sider whether it was deliberate or precipitate, what was the 
degree of evil intention, whether it was attempted only or 
perpetrated, what was the amount of harm done : the dis- 
tinction between the creations of art and the products of 
manufacture, or the relation of a free reproduction to a 
literal copy, presents similar ambiguities. To fix the limits 
of the conceptions is of more importance here, since by the 
operation of law certain advantages and disadvantages 
follow regularly and directly according as a given case is 
judged to belong to the one or the other ; but here also, 
though we take count of common usage, it is yet necessary 
in the main to distinguish them by positive enactment. 

183. Obviously we may set down any conception M as 
equivalent to any other conception N when we have by 
further specification so changed TV that it is equal to M. 
Thus there arise a number of incidental aspects or va- 
riations of the expression for the same M, which we shall 
further on find to be of use in enabling M to be subsumed 
now under this law and now under that, such law leading 
to a new assertion about M. There is no limit to the 
extent to which this procedure may be legitimately carried 
so long as the transformed M really coincides with the 
original Jlf 9 so long that is as TV is equal to M. We 
may even bring a triangle M under the conception of a 


four-sided figure N^ provided of course that we add triat 
one of the four sides is reduced to nothing. This may 
seem mere trifling, but it is useful in practice : we can 
thus for instance easily picture to ourselves how every 
time that two sides of a polygon, which were before sepa- 
rated by an intervening side, are made to meet at their 
extremities by the vanishing of the intervening side, the 
sum of the angles of the polygon (in this case four-sided) 
is diminished by two right angles. 

This use of transformation will engage our attention 
further on; what I wish here to emphasise is that the 
difference between the two conceptions thus brought to- 
gether is of course not altered by it. The four-sided 
figure remains just as distinct from the triangle as it ever 
was, i. e. so distinct that it must be stripped of its very 
essence before it can be ranked with the other; and 
similarly the alterations, whatever they be, that must be 
made in order to turn N into M, give the measure of 
the abiding difference between the two conceptions. When 
we are dealing, not as in this case with abstract con- 
structions of thought but with realities, which have an 
independent origin in the region of fact, such transfor- 
mations have very little value ; they are in the first instance 
mere fancies, whose significance cannot be ascertained 
without special enquiry. In thought we may change any 
given form of crystal into any other that we please by 
cutting off slices here and there, by successive alterations 
of outline we may change the likeness of a crocodile into 
that of a bird, from any one chemical element we may 
in thought derive all the others by giving successively 
certain other values to the coefficients which the funda- 
mental properties of matter take in the case of that one. 
But by such devices we cannot make the conceptions 
M and N approximate to one another, for their difference 
remains always as great as the number of steps that we 
must take to get from one to the other; neither can we 


thus establish between the actual things which exemplify 
these conceptions such a connexion that one might pass 
over into the other. For that it would be necessary to 
prove that the physical forces of the elements which build 
up an actual crystal of the form M are such as to make 
it possible for the same elements to be also in equilibrium 
when arranged in the form N \ or -that the concatenated 
system of forces which determines the structural type of 
the crocodile and maintains it in life may be so modified 
by other natural influences that the form of a bird may 
actually grow out of it, that in short the order of nature 
actually contains impulses which realise the changes which 
we may choose arbitrarily to make in thought or upon 
paper. We cannot but remember, though happily as an 
error which we have outgrown, the wild caprice with which 
not long ago people would derive a word in one language 
from any casual word in another, and call it etymology; 
at the present day people need to be warned against pro- 
ceeding in a similar way to satisfy the newly-awakened 
desire to conceive all the various kinds of organic beings 
as evolved from one other, all fixed specific differences 
being done away. But, whether Darwin has succeeded 
or not in his attempt, we must at any rate allow that he 
has taken the greatest pains to point out the real processes 
of nature by which the transformation of one organic form 
into another which we can conceive in thought may have 
been actually brought about. 


Schemes and Symbols. 

184. IN this chapter I shall continue to treat of the same 
subject as in the foregoing, but from a somewhat altered 
point of view. The extent and importance of the difference 
between several ideal contents can, we ascertained, be 
precisely determined only when we find ourselves able to 
compare several differences of the same kind, i. e. when the 
ideas to be compared themselves form series, whose 
members proceed according to a law that can be more or 
less exactly stated, and when moreover from the nature of 
the feeling whose modifications, distinct both in quantity 
and quality, are represented by the members of the series, 
such modification can only take place in one and the same 
direction. Compound conceptions whether of things or 
properties^ situations or events, by reason of the number of 
the characteristics or of the aspects which they include, 
may be altered in various directions ; one or some or all of 
these characteristics and of these aspects may run through 
all the various phases of which they are capable ; and again 
the bonds which connect them may pass through all the 
various degrees of laxity and strictness and all the changes 
of form to which they are by their nature liable. 

Now there is no reason why the value or the extent of 
the difference between two such compound conceptions M 
and N should not frequently be revealed to us by a direct 
impression with as much certainty as we need require in 



the case in question : if however a more accurate determina- 
tion were needed for scientific purposes, we should have 
first to determine the values of the various scales upon 
which the several alterations take place, and thence to 
determine the value of the total alteration which separates 
jft/from jfVor jA^from O. The reader may be inclined to 
object at once that in most cases at any rate we proceed in 
the reverse order to estimate the significance of the scale of 
a change which has taken place by the amount of the 
change which this alteration has produced in the total im- 
pression. I may allow this objection without taking any 
further notice of it ; for what I here wish to illustrate is not 
a logical rule but a propensity of our reason, which needs 
to be checked rather than to be indulged, but which as it is 
ineradicable needs to be specially mentioned. It is easy 
to understand, I mean, how out of the above-mentioned 
problem may arise the wish to have a universal scheme in 
which not only all the modifiable relations of different 
elements that we can think of, but also the values of 
the difference between any two modifications should be laid 
down so completely that the difference or the kinship 
between any two conceptions Mand ^should be exactly 
indicated by their position in the universal scheme. 

185. To illustrate this I will first go back to remote 
antiquity, to Pythagoras, To reconstruct a body of genuine 
Pythagorean philosophy out of the scanty and for the most 
part very questionable materials at our command is a task 
which I will not undertake, but I think I am able to state 
what may have been the fundamental idea which animated 
it, and which would enable us to understand why the 
sympathy stirred by it has been so lasting though often so 
perversely expressed. It is tolerably certain that the bent 
of the school was first to abstract mathematics, and 
secondly to their application to the processes of nature. 
The first line of study could not fail to lead them to picture 
the series of numbers and the world of shapes as two great 


coherent systems, and further to bring them to see how 
spatial figures themselves depend upon the numerical 
magnitudes which they involve. The second, besides other 
less known results, led to the discovery of the relation 
between the pitch of a note and the length of the vibrating 
string, and thereby no doubt suggested the general idea 
that even phenomena whose differences are in the first 
instance felt by us as differences of quality are based upon 
mathematical differences that admit of comparison. The 
rash generalisation of results thus won is what the fancy of 
men is always prone to ; the mathematically-trained Pytha- 
gorean went so far as to make the reflexion that if it be 
once established that a scries of changes in phenomena 
corresponds to a series of changes in magnitude, then every 
other conceivable mathematical relation along with all its 
modifications must have its counterpart in the phenomena, 
or conversely, if a group of phenomena is based upon 
definite relations of magnitude, the coherence of all the 
processes of nature necessitates the conclusion that all other 
phenomena also depend in like manner upon relations that 
can be mathematically determined. 

This I conceive to have been the origin of those specula- 
tions which Aristotle expresses by saying that Pythagoras 
regarded the principles of numbers as the principles of 
things : but we must further consider the meaning of this 
expression. The purport of the Pythagorean philosophy 
was certainly wider than we might be led to suppose by 
that other saying of its author, that God has ordered every- 
thing by measure and number; i.e. it was not limited to the 
mere application of mathematics to nature, if that means 
merely that the definite magnitudes of natural forces and 
processes modify one another when brought into contact 
according to the same mathematical laws that hold good 
for magnitudes in general : these data themselves, to which 
mathematics are merely applied by modern * mathematical 
physics/ were regarded by Pythagoras as in themselves form- 


ing a system whose inner articulation is based upon the same 
relations that determine the structure of the series of 
numbers and of all their possible combinations. I wish to 
distinguish in this theory a general idea and the particular 
form given to it. 

186. The so-called natural philosophy of the lonians had 
devoted itself to describing the processes by which natural 
bodies were formed out of their primitive matter and 
returned to it again/ As this philosophy very generally 
used for this purpose the ideas of condensation and 
rarefaction, it may appear, in virtue of its employment of 
quantitatively determined means, to be closely akin to the 
Pythagorean theory. The two are nevertheless very far 
apart : for the lonians never betray any desire to show that 
the sum of that which is thus produced at any moment of 
its existence or in the whole series of steps by which it 
comes into existence forms a coherent whole of mutually 
dependent parts. Pythagoras on the other hand seems to 
have troubled himself very little about this origin of the 
world, but the world as it was after it had come into 
existence was to him a system, such that not merely were 
its parts there, one beside the other, but that there would 
have been a gap in it if while one phenomenon were 
present another had been absent. If a and b and d are 
present, then if c is there at all, it is not merely there along 
with the others, but it is there because the law according 
to which the series a b advances to d requires it as the third 
member of the series which is indispensable to the presence 
of the fourth member d : or if c is absent, it is not merely 
absent as a matter of fact, but because the law which 
regulates the series excludes the possibility of this third 
member before d. The same consideration may be applied 
to other series in the actual world, to a /3 y 6 and to a fc c ft, 
and this application was made by the Pythagorean school. 

How they conceived the relation between the different 
characters of these series, which I wished to indicate by the 



use of different alphabets, is a point upon which we are 
certainly in the dark, and upon which, as we may gather 
from Aristotle, the fullest information would probably throw 
but little light ; but with respect to the law which in each of 
these series binds the homogeneous members together, 
it seems to be indubitable that it was regarded as precisely 
identical for all the series, i.e. that they maintained a 
complete parallelism between the relations prevailing in the 
various groups of connected phenomena. This is shown in 
the supposition that the earth has an invisible fellow, in 
order to bring the total of the then known planets up to ten, 
to which number the arithmetical mysticism of the system 
had once for all assigned a peculiar significance, in the 
assumption of a fifth element, which together with water, 
earth, fire, and air, shall correspond to the five regular solids, 
tetrahedron, cube, octahedron, dodecahedron, eicosahedron, 
in the attempt again to conceive the distances of the 
planets as arranged according to musical intervals, and 
even in the meagre form of their tables of opposites. To 
us of course these tables do but illustrate the frequent 
occurrence of this relation of opposition between two con- 
ceptions even when these are arbitrarily chosen, but the fact 
that they always contain ten pairs seems to indicate that 
they were intended to represent this relation as essential 
for all ttje different stages in a series of ten members. 
Finally when they assigned life to the number six, intelli- 
gence and light to seven, and friendship to eight, we see 
that they regarded not merely the phenomena of nature, 
but also those of mind, and in a word every conceivable 
thing, as ordered according to the same serial law. 

This philosophy then sought and fancied that it found 
precisely what we spoke of above, viz. a universal scheme 
which mounting from simple to complex was supposed to 
embrace the whole sum of possible forms, one of which was 
to serve as a pattern for the formation of every actual thing, 
while at the same time these forms or types were to be so 


arranged in the scheme that the position of its type directly 
determined the significance of every actual thing, and the 
amount of the difference or the kinship between it and 
other things formed upon the model of other members of 
the series. The general idea then that I would ascribe to 
the Pythagorean philosophy is this, viz. not merely a subse- 
quent arrangement of things whose nature was originally 
settled without reference to the principle of this arrangement, 
but a harmony of the Cosmos which name was first applied 
to the world by Pythagoras based upon the notion that all 
things are from the beginning nothing but various realisa- 
tions of a series of types, regulated by one law of development 
which is the same for all. 

187, The general conception is undeniably grand, but 
grandeur is sadly lacking in the special form here given to 
it. Even in the present state of the mathematical sciences, 
various as are the magnitudes whose interesting mutual 
relations have been examined, it would be impossible to 
find adequate types or symbols or abstract expressions for 
the still more various relations that subsist between the 
elements of the actual world and the combinations that 
arise out of them ; but the arithmetic of the ancients, which 
the Pythagorean school seems to have helped to develop, 
furnished in its then state but very few and very meagre 
numerical relations, whose significance must have been 
much exaggerated and from the beginning very arbitrarily 
interpreted before they could be regarded as the relations 
upon which the structure of the world is based. The 
grounds on which they justified their well-known veneration 
for the number ten, viz. the fact that all numbers are 
generated by the repetition of unity; that in this series 
the even numbers alternate with the odd numbers, which 
cannot be divided by the principle of multiplicity,' i.e. by 
two, and which are therefore held to be of higher rank; 
that three is the first union of odd and even, four the first 
square of a multiple number, and ten the sum of these 


exalted four first numbers, are grounds which could not 
be admitted except by a system of symbolism which was 
ready to accept any interesting motive without regard to 
its connexion with others : though the real grounds of that 
veneration undoubtedly lay in the habitual use of the 
decimal system. If these thinkers had been acquainted 
with all the algebraical and transcendent forms of functions 
which are the instruments of modern mathematicians, how 
much more various would have been the symbols employed, 
and how much more delicately would they have been 
adapted to the nature of the several phenomena ! The 
same tendency still survives in us : even in cases where 
calculation in the strict sense is impossible we are inclined 
to use the term 'power' 1 when the meaning and importance 
of a conception is raised in some peculiar manner, as for 
instance when each of the centres of relation, whose deter- 
mination by each other constitutes the meaning of the 
conception, is itself exalted into a small system, whose 
members determine each other in the same way. 

We can imagine then how the Pythagoreans (if they had 
had our knowledge) might have illustrated many relations 
of dependence between various elements by the relation 
of a logarithm to its number, and how they might have 
applied trigonometrical functions to explain any kind of 
periodicity. As however they had not our resources at 
command, and as even these would still be insufficient, it 
would be quite useless to examine in detail the reasonable- 
ness of the Pythagorean symbols. 

188. That it was the fate of the whole theory to be 
variously interpreted and misunderstood is easily explained 
by its nature. According to one statement of Aristotle it 
was the principles of numbers that Pythagoras identified 
with the principles of things. This seems quite intelligible. 
By these principles of numbers must be meant the relations 
between one and the other numbers, the way in which one 
1 [In the mathematical sense.] 


can be repeated, the divisibility or indivisibility of the rest, 
in a word the possibility of generating the whole series 
of numbers by the use of these constant relations and 
operations, or, as we should say, the possibility of exhibiting 
every number as a function of other numbers. Things 
then, ought also to have the same inner structure, their 
series ought also to be arranged according to the same 
principles, so that the nature of the one might be exhibited 
as a function of the nature of the other. 

But it is also asserted by Aristotle along with others that 
the Pythagorean school declared that numbers were things, 
or at any rate that things were numbers. Even this is quite 
intelligible to any one who is acquainted with the history of 
philosophic ideas and the customary ways of expressing 
them. To a certain extent indeed the Pythagoreans would 
have been right in making .this assertion, and this justifies 
us in supposing that they actually made it ; for as already 
said what they intended was by no means merely to apply 
numbers to the quantitative determinations of things whose 
real nature is independent of these determinations, e.g. you 
may have similar triangles of very various sizes : their 
numbers were meant to signify that which distinguishes the 
essential character of one thing from the essential character 
of another ; a was a because its content was constructed 
according to a the function-form or the generating law of 
one symbolic number, and was thereby distinguished from b 
which was b because it followed j3 the generating law of 
another symbolic number. It was quite possible then to 
say, with a reservation to be presently noticed, that the 
essence of a thing, in the sense of that which distinguishes 
it from another thing, lies in the number immanent in it 

The other assertion that the essence of things, in the 
sense of that in virtue of which they all are things, or their 
reality, consists in these numbers, or that numbers are the 
real things, was perhaps not positively made by the Pytha- 
goreans in this form : if they did make it, they certainly 


could not justify the latter expression, but they could as- 
suredly justify the former : for if there is actually nothing 
whose nature is not determined by one of these symbolic 
numbers, the numbers are assuredly the conditio sine qua non 
of every reality; to treat them as more than this, and to 
speak of the numbers themselves as the real things, is an 
unwarrantable straining of language, though we shall pre- 
sently see how prone to it the thinkers of all ages have 

There remains one great imperfection which we have 
already mentioned. The same typical series of numbers 
has to repeat itself in a number of parallel series of actual 
things, m a l> c d, /3y6, a b c & ; how then are the mem- 
bers b ft b distinguished from one another if the whole 
nature of each of them is exhausted by the same symbolic 
number? To this there is no answer possible: at this 
point the theory, which aimed at embracing the nature of 
things completely, relapses again into a mere application 
of a general law of structure to various cases whose charac- 
teristic differences must be regarded as given. But this is 
what makes it serviceable for our present purpose as an 
illustration; it thereby becomes an attempt to frame a 
universal scheme for the relations of kinship and 
difference between all the groups formed by kinds of 
content that can ever by any possibility come to be con- 

189. In order to justify the length of this discussion I 
would point to the extraordinary tenacity with which this 
desire to find a scheme for the whole contents of thought 
has maintained itself through the course of ages. It showed 
itself first in this form of mystical speculations about 
numbers; over these we may pass very lightly; as such 
speculators were satisfied with anything however meaningless 
so long as it was interesting and startling, they were, to 
speak plainly, always in search of a secret truth which they 
never found, and it must always have needed a very sym- 



pathetic hearer to find in the symbols a better expression 
for the meaning put into them than could have been 
obtained without them. 

Presently the speculators ceased to found their dreams 
on this purely arithmetical basis and wandered away in 
various directions. In the first place every discovery made 
by advancing science that has any important bearing upon 
the relations of things has almost without exception been 
extended into a scheme for the articulation of the whole 
world. For a long time people traced everywhere the 
behaviour of the four elements of the ancients ; and in 
later days the mystic significance of this number four did 
not pass away, it was only transferred to the newly dis- 
covered constituents of organised bodies, 'carbon, hydrogen, 
oxygen, and nitrogen; it agreed admirably with the four 
quarters of heaven, for zenith and nadir of course fall 
outside our natural line of sight ; it agreed equally well 
with the four seasons of the temperate zones, within which 
these speculations were carried on, and with the four in- 
dispensable cases of nouns ; at a later date, as the theory 
of astronomy came to completion, the contrast between 
centrifugal and centripetal tendencies entered into men's 
notions of all things and was fused into one with the 
opposition of the sexes and the relation of acid to alkali ; 
the discovery of magnetism and electricity caused the 
scheme of polarity to be carried even further if possible 
into the consideration of all conceivable things. 

Other speculators proceeded in the opposite direction, 
starting from the just reflexion that even the relations of 
numbers are, in part at least, only instances of other still 
more abstract fundamental relations ; these then (they hold) 
must be sought, and will be found if we simply reflect upon 
the operations by which our intellect does in fact arrive at 
its ideas of all things whatever. Now every idea, or at least 
every compound idea, is made by setting down an a, dis- 
tinguishing from it or opposing to it a b, and finally bringing 


both into a relation c thus thesis, antithesis, and synthesis 
come to be regarded as the scheme upon which all reality 
is constructed and as the rhythm which thought must main- 
tain in the orderly consideration of that reality. But it is 
easy to see that the more abstractly these symbols are con- 
ceived the more they pass over into notiones communes 
which do indeed apply pretty well to everything but give 
us no adequate knowledge about anything. Logic then 
meets all this wild talk with the demand that things be 
considered, divided, and investigated simply and solely 
with reference to their several natures, for there is no 
universal scheme that can be applied, and the employment 
of merely fanciful models can only injure the impartial 
quest of truth. 

19O. Of this unfavourable verdict I can abate nothing, 
and in some remarks which 1 wish still to add I have no 
such intention. When the content M of a conception, an 
idea, or a perception is given to us in such a manner as to 
unite in the form ju, a number of characteristics, or parts, or 
points of relation, it is a quite justifiable scientific curiosity 
that prompts us to enquire how the examples of M will 
behave, how they will be altered and distinguished from 
one another, when we vary within the allowable limits either 
the parts of M only, or both them and the general form of 
union \i. 

In the first place if we keep to the former kind of altera- 
tion, there will usually be but little interest in tracing all 
the kinds of M that are got by simply changing the quantity 
of the characteristics, for these kinds will, in most cases at 
least, resemble one other and only repeat the same thing 
on a different scale. But if one of these characteristics m 
be of such a nature that for it the opposition of negative 
and positive has a plain and palpable meaning (such an 
opposition for instance as there is between right and left, 
attraction and repulsion, concave and convex, and generally 
between ascent above a zero-point and descent below it) 

S 2 



then it concerns us greatly to know what happens to M 
when we substitute m for + m in its generating law. 
Supposing y ~ f x is the equation of a curve, we always 
take the trouble to set down in turn the positive and 
negative values of x, and not till we have united the results 
thus obtained do we think we have arrived at the nature of 
the curve, which in this case presents itself to our per- 
ception not as a mere generality, but as the whole which is 
got by combining every possible example of the general 
equation. If we happen to see, in a piece of ornamenta- 
tion, a volute which bends downwards to the right, our 
imagination is stimulated in a similar way ; even if we have 
no mathematical knowledge of the generating law of this 
curve, we understand, by reason of the homogeneousness 
of directions in space, that the volute might be repeated 
in a precisely similar though opposite bend upwards to the 
right, and again with another opposition upwards to the 
left and downwards to the left. If now these continuations, 
suggested by the beginning which we see, are not carried 
out, though the surroundings do not give any obvious 
reason for this incompleteness, our aesthetic feelings are 
unsatisfied, but this demand for symmetry has also a 
logical foundation. It is of the very essence of a law that 
it shall apply to all variations of the points of relation which 
it comprehends ; there is therefore a contradiction in a 
perception which suggests a law together with the possibility 
of its prevailing universally, and yet actually presents it as 
prevailing only in part : what we miss in the perception 
appears as a defect in the thing : we supply it in order to 
remove the groundless want of universality. 

We always feel a similar impulse in examining con- 
ceptions. Whenever in any M one of its determinants 
may vary from -f m to m, which it can only do by 
passing through the intermediate value m = o, the tripartite 
division thus suggested becomes for us a scheme^ which we 
take as the basis of our investigation of the whole extent 



of M. This is the point which I wish here to emphasize, 
in order to mark the difference between this proceeding and 
the wild dreams we have just condemned, viz. that this 
scheme can be nothing but an invitation to turn our enquiry 
in a particular direction^ and cannot give us by anticipation 
a picture of the result at which we shall arrive. It does not 
always happen, as in the case of the volute, that the 
counterparts we expect can be found : whether the change 
from + m to m gives other possible kinds of M at all 
depends upon the nature of the form of union /x. Still less 
can we see beforehand whether the kinds thus obtained will 
be in any way proportional to the differences of the con- 
ditions, and if so in what way : it is quite possible that for 
a certain JJL this absolute opposition of + M and m is 
absolutely meaningless. Our method then will be to let ju, 
likewise pass through all the possible forms given by the 
various alternatives ; here also for mere additions of quantity 
we shall expect only a series of similar results, but for every 
cardinal point at which p, takes a qualitatively different 
significance or passes at a bound into its opposite we shall 
expect a quite new formation to appear in M which depends 
upon /x ; and lastly for every remarkable feature which we 
find in a special case of M we shall expect to find as 
counterpart an equally remarkable feature in a similarly 
conditioned special case of a similarly constructed N (as 
for instance when we find that waves of light behave in a 
certain way we look for corresponding behaviour in the 
waves of sound) : but all this remains only a question put 
to the object, to which we await the answer : the answer 
which enquiry yields may turn out quite contrary to what 
we expect, but must be accepted whatever it be. Where 
those dreamers deceived themselves was in supposing that 
whenever their scheme which they assumed to be universal 
was applied to any matter whatsoever, every place in it 
would always be filled by some remarkable form of that 
matter, none would ever remain empty, and further in 


supposing that as these various matters, passing through 

the same sequence of changes, filled up the several places 
of the scheme, the forms which filled the same places would 
by a striking resemblance or analogy in their whole character 
announce themselves as connected, as akin to or as coun- 
terparts of one another. When this was not the case, there 
was a strong temptation to try to fill up the gaps by ground- 
less suppositions, and to restore the desired symmetry in 
the corresponding members by giving undue prominence 
to secondary features. 

191. Among modern attempts to unfold in a scheme the 
meaning of the world there have been some grand ones 
which even seemed to avoid an essential fault of the Pytha- 
gorean theory. In another work (' Geschichte der Aesthetik 
in Deutschland/ p. 176 ff.) I have examined at length the 
motives which led to the development of the Hegelian dia- 
lectic, the most important of these attempts ; I will content 
myself here with making a few remarks on its logical char- 
acter. The Pythagoreans in conceiving development in 
countless parallel series with different contents took no 
count of the differences by which the corresponding mem- 
bers of the various series are separated from one another in 
spite of their occupying the same place in the general 
scheme. The decimal system, with its ascending powers of 
the number ten, never led them, as it might w^ll have 
done, to treat these parallel series as themselves successive 
periods of one and the same main series, resembling 
one another in their internal structure, but raised one 
above the other so to speak by the height of the level 
at which they exhibit this structure, like the octaves in the 
musical scale. 

The imagination of the modern philosopher has supplied 
this deficiency; the many parallel series are contracted into 
a single series, composed of cycles of similar structure, the 
last member of each cycle making a starting-point of a dis- 
tinctively new character for the development of the next. 


If it is possible to find the first member of the whole series 
and the law which determines the form of the first cycle, the 
variety of the contents which form the members of the fol- 
lowing periods may be explained by their distance from the 
starting-point and the transformation which the initial mem- 
ber has undergone at each step of the way. Hegel then 
requires us to concede as a metaphysical presupposition, of 
whose correctness logic cannot judge, that the world is no 
sum of things that stand and events that go on one beside 
the other, the former standing quiet till they are stirred to 
change by a stimulus from without, the latter determined in 
their inter-action and in their whole course by universal 
laws that hold good always, but that instead of this all the 
variety of the world is only the development of a unity that 
never rests, all events only stages in this development or 
secondary effects of it, and things themselves but appear- 
ances, either transitory or begotten anew at every moment, 
whose whole being lies in the active movements of that 
unity, crossing each other and coming to a focus in them as 
subordinate vehicles of that development. 

In this account of Hegel's point of view I make no pre- 
tence to unimpeachable accuracy, which it would be difficult 
to attain in a long exposition and quite impossible in a short 
statement ; but what has been said is enough to enable us 
to understand that within each dialectic cycle these different 
forms, whose significance somehow constantly increases, 
cannot simply occur one beside the other, but that each 
must issue out of the preceding one : development, in short, 
is the very essence of the system. 

102. Now no development is imaginable without a defi- 
nite direction which it takes in contrast to others which it 
does not take ; but it is equally clear that in. this case above 
all others it is impossible for the unity which develops itself 
to receive this direction from without; it must be determined 
by the nature of that unity itself. But here we find that no 
accurate and exhaustive expression can be obtained for the 


entire nature of that which under the name of the absolute 
is regarded as the one basis of the world, but that what we 
mean by it in a sort of presentiment is fully revealed to us, 
nay comes to be completely itself only in and through the 
development, indeed, the very name indicates this, for as 
it is nothing but development, it cannot be itself before it 
has begun to develop. 

The only point of departure then that is left for us is this 
fact itself, i.e. the knowledge that the absolute is not rest 
but development. Assuredly then its development must 
take that direction and form which follow from the concep- 
tion of development itself, and which therefore must recur 
in every example of the conception. This opens up a very 
simple line of thought. If any A is to develop itself, it 
cannot already be that into which it has yet to expand itself ; 
neither can it not be, or be void of content, for then it 
would not be the determining ground of that which is to 
be ; as yet unexpandcd and shapeless it must still be the 
determinate possibility of its future growth, in a word it 
must be ' in itself ^ ' or potentially that which it is to become. 
But its nature would not consist in development if it were 
to abide in this potential state ; it must actually become 
that which it is its nature to be able to become. But 
becoming or the process of development is only an inter- 
mediate step between possibility and fulfilment ; as merely 
coming to be, hovering between starting-point and goal, 
that which is developing itself would be neither identical 
with itself as it was in its potentiality, nor yet already that 
which it has to become. This at once enables us to see 
why the second stage of the development, in which that 
from which we started is as it were divided against itself, 
was called by Hegel ' other being ' or ' being otherwise 2 ' ; 
we see it still more clearly when we remember that it is to 
the ground of the whole universe that this unfolding is in 
strictness ascribed ; the process of its becoming does not 
1 [' An sioh.'] 2 [ Anderssein.'] 


consist in a simple movement in a straight line, but in the 
generation of an infinite variety of forms, of which it was 
the possibility; each of these is one of its results, none ex- 
presses its whole nature ; the sum of all may indeed contain 
a complete expression of this whole nature, but only for the 
observer who adds up the sum and combines this manifold 
into a unity in his thought. But that which is developing 
itself must be this unity not only for others but for itself, if 
it is actually to become that which it was its nature to be- 
come ; and thus the name of t being for self l ' is given to 
this third stage of the cycle, signifying the completion of 
becoming, the attainment of the end of development, the 
return of the potentiality into itself. This return of course 
is not a simple return ; i.e. we do not mean that the inter- 
mediate stage of the process is set aside 2 without leaving 
any result behind or wiped clean out ; it must be set aside 
in the sense of being stored up and preserved; the last 
stage, being for self, is richer than the first, the potentiality, 
by the history of the process through which it has come 
into being. 

It is easy to find images for this; thus the octave of 
the initial note is a return of the latter into itself, and 
yet preserves in its heightened pitch the result of the in- 
tervals through which it has passed ; thus when a mind, 
in which universal truths were innate in the form of 
methods which its thought instinctively followed, had, by 
passing through various experiences and enquiries, in- 
volving doubt and the removal of doubt, arrived at a full 
consciousness of these truths, it would merely have returned 
to itself and yet would be enriched. I will forbear however 
to explain in detail the peculiar meaning of these phrases ; 
for us it is enough that in the third stage of the develop- 
ment something is given which is indeed a consequence 
of the first stage, yet is not identical with it but opposed 
to it as actuality to possibility. 

1 [< Fursichsein.'] 2 [' Aufgehoben.'] 


Thus understood the three moments or stages of ' being 
in itself,' ' other-being,' and c being for itself,' are but the 
component parts of the conception of development, and we 
shall be able to recognise them in everything that develops 
itself. But Hegel's system rests, as we said, on the con- 
viction that the whole content of the universe, the whole 
intelligible world, i. e. both nature and mind, are but stages 
in the development of the one absolute, and that within 
each of these great provinces the several members proceed 
in the same rhythmic order, each founded upon and issuing 
out of that which goes before, and that accordingly the sum 
of all that is intelligible and all that is real would present 
itself to us if we knew it completely as a great series, whose 
several periods are similarly constructed but have each 
a peculiar significance in its content which is ever rising 
higher and higher. Upon this conviction we do not here 
intend to pronounce any judgment ; but it remains for us 
to ask what is the logical value of the dialectic method just 

193. It is easy to see that it is not strictly speaking 
a method in the sense of a direction how to find something 
that we are in search of; it is rather a scheme^ in the sense 
in which we have used the word above, which only invites 
us to enquire if anything is to be found in a given direction 
or in a spot already marked out, and if so what it is, though 
of course it implies a confident expectation that the search 
can never be in vain. If we try to apply this scheme to the 
independent treatment of a generic conception M, in order 
to arrange its various species in a series corresponding 
to their essential resemblances' and differences, or if we try 
by means of it to exhibit in their true relations to one 
another a series of conceptions which are connected by 
a variety of other circumstances (as e. g. right, wrong, crime, 
and punishment are connected), we at once find how un- 
certain it leaves us as to the direction in which our thoughts 
are to be turned. It is possible that this uncertainty might 


vanish if we could appeal to a complete philosophy which 
had already set down in a universal series the history of 
the development of all that is thinkable, and had therefore 
arrived at a conception of right so perfect as to reveal 
at once the direction of its further dialectical development. 
But to say this would be to deny from the beginning the 
applicability of the method as a universal direction for the 
discovery of truth ; it can prove itself such only by this 
independent service which we require ; i. e. it must be able 
merely by means of its form of procedure to teach us how 
to develop any given conception in all its proper con- 

Suppose then that we have given us the general concep- 
tion of right, for evidently the other three that we named 
refer to this as a primary conception already fixed : what 
now is it 'in itself or potentially? into what 'other- 
being ' does it pass over ? into what * being for self ' does it 
return ? It is at any rate evident that a right involves 
an estimate of relations which prevail between the claims of 
various persons to exercise their wills upon some object 
which brings them into collision. It follows that there can 
be no right if there be no world with relations and objects 
for the exercise of will, or if there be no persons who can 
direct their wills to the same ends in one and the same 
world. Right then is only potentially right and not yet 
that which according to its conception it is to be, so long as 
it only denotes by anticipation the approval or disapproval 
of relations which do not yet exist. 

Its * other-being ' is also quite intelligible ; it all comes to 
the simple truth that general conceptions mean nothing 
when there are no particulars for them to connect ; the 
( other-being ' of right consists in the various rights whose 
conditions lie in the existence of this nature, of these human 
personalities with these definite wants and claims ; after the 
general doctrine which sets forth the conception of right 
will come the special doctrine which contains its applica- 


tions. This direction is so simple that we do not need 
to wait for the dialectic method to teach it to us ; but that 
method does not help us in the least to carry it out ; for 
after all experience alone can teach us what conditions 
do in fact exist which give occasion for the development 
of the general idea of right into special forms of right. 

194. There is, however, yet another kind of advance that 
we can conceive. ' Other-being ' certainly does often mean 
the passing of the universal into its various particular 
forms ; but I have already remarked that the Hegelian 
doctrine lays stress upon the relation of opposition which 
prevails between the two members, including the opposition 
of the universal to the particular : this idea of opposition, 
universalized and carried to its extreme pitch in the concep- 
tion of contradiction, gives a further meaning to other- 
being,' it may stand for the simple contrary of that which 
the first (the being in itself) stands for. In pursuance 
of this train of thought, right was made to pass into wrong ; 
and wrong was made to issue in punishment, not indeed as 
the ' being-for itself/ but as the means of reasserting the 
violated right by the negation of its ' other-being/ i. e. of 
the crime. 

Now here again we have nothing that would not be just 
as clear by itself without all this apparatus of the dialectic 
method; and further, the method is actually confusing. 
Any unprejudiced person would say to himself on reflexion 
that all right has living reality only when living persons not 
only know it but respect it in their actions, but that the 
movements of men's wills are not in fact governed by 
the ideal which they ought to follow ; wrong and crime 
therefore appear, not as something necessary that must 
exist, but as something possible that may y and indeed 
always will^ exist, to judge by what experience teaches 
us of human nature. In the transition which the dialectic 
method gives there is none of this cautious bridging of 
the gap between the two conceptions; it is represented 


as part of the very conception of right that it shall pass 
over into wrong, and the paradox is not to be justified 
by a plea which will be presently considered. 

The transition to punishment as the third stage offends 
us less merely because we supply the motives which are in 
truth not given at all by the method itself. The method 
does indeed demand restoration of the right, and that 
by negating its negation the wrong; but it does not tell 
us by what procedure this task, stated abstractly as the 
negation of the wrong, is to be carried out. Why should it 
take the shape of punishment? The evil disposition out 
of which the wrong sprang is equally negated by disapproval 
and by improvement, the harm done by payment of 
damages, the violation of the dignity of the law by re- 
pentance, and by a fresh recognition of its bindingness. 
All these considerations show that the dialectic method was 
of no use here except as a scheme, with places marked out 
which we might seek to fill, but that, though we were 
tolerably successful in filling them, the content with which 
they were to be filled was only to be got from a quite 
independent examination of the peculiar nature of the 
object in question. 

195. We said that it seemed to us absurd to maintain 
that it is part of the very conception of right to pass over 
into wrong ; but this swinging round of a conception into its 
opposite has been so often and so emphatically claimed as a 
higher truth discovered by dialectic, that it is worth while to 
return to the point. Hegel remarks 1 that at first of course 
the understanding fancies it can apprehend the nature and 
truth of the real world by a number of fixed conceptions 
complete in themselves and exclusive of each other; but 
that the truth is that different conceptions do not simply 
stand one beside the other with equal claims to represent 
the finite, but that the finite of its own nature does away 
with itself, and passes over of itself into its contrary. Thus 
1 [Vol. VI. of his collected works, p. 152 f.] 



we say that man is mortal, regarding death as something 
whose ground lies merely in external circumstances ; and 
according to this view man would have two distinct pro- 
perties, that of living and that of being mortal also. But, 
according to Hegel, the true way of regarding the matter is 
that life as such contains the germ of death, and that in a 
word the finite in itself contradicts, and thereby does away 
with itself. 

Here we can detect, more readily than we can in some of 
the other passages in which Hegel treats of dialectic, a 
confusion between two different statements. It is to the 
conceptions by which we try to apprehend reality that fixity 
and completeness are attributed in the first sentence : it is 
not the conceptions but the finite thing to which we apply 
them that is said to pass over into its contrary, and in 
this latter statement lies all the truth that the passage 
contains, which truth is shown by what follows to have been 
uttered unintentionally or even contrary to the intention of 
the author. For when the finite as such does away with 
itself, it does so not because the general conceptions which 
apply to it have lost their definiteness and swung round into 
their contraries, but because it, the thing to which those 
conceptions are applied, as finite or as actual, is unable 
permanently to fulfil what is required of it by these concep- 
tions, though each of them is true of it at one , moment ; 
through a defect in its nature it passes out of the province 
of one unchanged conception into the province of another 
which is equally unchanged. But the conceptions them- 
selves do not alter their eternal meaning because it is only 
for one moment perhaps that they are a correct measure of 
the changeable objects to which they are applied. 

The true view of the matter then cannot be that life as 
such bears in it the germ of death, and that the finite in 
general contradicts itself: it is rather the two parts of this 
statement that contradict each other. Life as such does 
not die, and the general conception of life obliges the living 


thing to live, not to die ; it is only the finite, mentioned in 
the second part of the statement, i.e. only particular living 
bodies that carry in them the germ of death. And even 
they do so not in virtue of the idea of life which is realised 
in them, but assuredly only by force of external circum- 
stances, i.e. only because that combination of material 
elements through which alone life is manifested on the 
surface of this earth is unable to exhibit an undying 
example of life, though that would in no way contradict the 
idea of life, whether this inability be regarded merely as 
a result of the laws of nature which are here in operation, or 
as part of a universal plan. 

Similarly right never itself passes over into wrong, but 
sometimes the will of a living person which ought to 
embody it may, through want of judgment or through the 
impulse of passion, be led into wrong while striving to do 
right, and sometimes the law, which, men being what they 
are, could not be administered at all if it allowed exceptions, 
may do a wrong in a particular case involving complications 
for which no provision has been made. 

Logic then can in no way accept this doctrine that con- 
ceptions dialectically do away with themselves : but the real 
world as we find it is so arranged and ordered that what is, 
though it does not do away with itself, yet docs of its own 
nature pass from the province of one conception into that 
of another ; and the fact that we find it so is worth notice, 
as a fact about things that is to say, not as a peculiarity of 
the intellectual tools by which we come to know them. 

196. In any case, even apart from all the objections here 
raised, the dialectic method would in the end give us only 
an arrangement of our conceptions, an arrangement which 
might no doubt present various points of interest to persons 
fond of reflecting and comparing, in the aesthetic impression 
produced by the discovery of analogies, parallels, and 
contrasts, but which would scarcely open up a new way of 
knowledge that could lead to definite new judgments or 


propositions, or to a better and more precise settlement of 
questions hitherto doubtful. To supply this want which the 
dialectic method fails to supply is precisely the aim of other 
vast attempts, viz. the attempts to found a logical language, 
a universal mode of characterising conceptions, or a philo- 
sophical calculus, at which Leibnitz laboured so long. The 
mere addition of a series of large numbers would be an end- 
less task if we were obliged to have a distinct image of each 
one of the thousands or hundreds of units composing them, 
and to build up each of these numbers separately and at 
last their sum by repeatedly adding unit to unit. But our 
system of ciphering enables us, without the need of 
distinctly forming even any collective idea of the numbers, 
to set units under units, tens under tens, hundreds under 
hundreds, and then, by adding up each of these simple 
columns, unerringly to bring out a result which itself in turn 
we are quite unable to represent adequately in a single 
picture by any effort of our imagination. 

Now our conceptions so far resemble numbers that they 
also contain for the most part a variety of individual images, 
whose union with each other is not distinctly before us at 
every moment, but only thought of in one collective im- 
pression ; but they are denoted by words far less perfectly 
than numbers are by figures. By the use of words that are 
akin (though we are often no longer conscious of- the fact) 
speech does indicate the kinship between contents, but very 
imperfectly, for kindred ideas are also denoted by inde- 
pendent roots : the kind of kinship between them is no 
less imperfectly expressed, for the small variety of ways in 
which derivatives may be formed is quite inadequate to the 
manifold relations that have to be indicated ; moreover, 
instances of each relation occur which, as the first to take 
the fancy of the framers of language, are denoted by simple 
words in which the characteristic derivative form is wanting ; 
and finally the name of a conception never gives us all the 
ideas that make up its content marked by simple signs and 



united in such a way that when we have to combine several 
conceptions M N O we may shut our eyes to the meaning 
of the whole and apply ourselves to combine some of the 
component ideas with the same certainty of arriving at new 
and correct results that our system of ciphering gives us in 
numerical calculations. 

These defects of language then we are called upon to try 
to amend ; we are to dissect all our conceptions till we have 
found the simple primitive ideas of various kinds which 
admit no further analysis and the simplest ways in which 
they can be combined, and we are to characterise these 
by fixed signs, in order to obtain by their combination a 
symbol for each conception which shall adequately express 
its content. We need not think that the object of this 
undertaking is the formation of a new speakable language, 
which could never supplant the national and historical 
forms of speech : its result would be a collection of for- 
mulae for the purposes of scientific thinking only, to which 
recourse might always be had for the settlement of the 
doubts which arise from the employment of ambiguous ex- 
pressions : for Leibnitz flatters himself that if we once got 
such an instrument disputants would always cut their 
quarrel short by an amicable agreement ' Let us reckon 
it out.' 

197. This is no doubt one of those enterprises whose 
execution alone can finally decide whether they are prac- 
ticable ; it would be over-hasty to deny the possibility of 
that which might after all perhaps be realised by a happy 
invention. However, the utter want of success hitherto 
makes the inherent difficulties of the task more evident for 
the present than the possibility of overcoming them. If all 
we had to do were to make a system of signs for marking 
the contents of our conceptions, the problem might appear 
difficult but not insoluble. For then we should probably 
begin by passing over all the generic conceptions of natural 
history and limit ourselves to those conceptions whose 




union in thought leads to difficulties which impede science 
or the practical deliberations of life. Nevertheless even 
this problem is harder than it seems, and the possibility of 
solving it derives only an apparent confirmation from the 
mathematician's language of signs and the symbols of 

It is characteristic of the mathematician that he reckons 
only with comparable elements, with magnitudes, the sim- 
plest combinations of which he certainly can symbolise 
quite clearly and unambiguously; but as the functions and 
equations thus obtained grow more and more complex, we 
see more and more plainly even here a sort of deterioration 
in their employment. In the place of denominations which 
really exhibit the inner structure of the magnitude in ques- 
tion so as to indicate quite plainly how they are to be 
treated in the calculation, we find introduced in order to 
secure the necessary conciseness arbitrary symbols which 
no longer have this property, but resemble the words of 
ordinary language whose meaning must be known quite 
independently of their sound. The expression V i still 
expresses the origin of the function for which it stands, and 
from this we can determine by general rules what results 
when we multiply it once or several times by itself: but 
this expression has already been discarded as too lengthy 
and replaced by the other expression / which asc it stands 
gives no clue to its signification, and whose meaning must 
be otherwise already known if it is to be used correctly. 
When we go on to speak of B-functions and r-functions, 
these expressions are certainly concise, but we can only 
understand them by representing them as equivalent to 
other lengthy formulae, which in turn are only made in- 
telligible by a previous explanation of the meaning to be 
attached to the general signs of magnitude and symbols of 
combination employed in them. All this is no reproach to 
mathematics, nor is it any proof of the impossibility of a 
universal system for characterising conceptions; it only 


shows that any formulae that the latter could give us would 
not by themselves tell us all we need know, but would pre- 
suppose a great deal which we should have to learn before 
we could even understand them. 

The symbols of chemistry make this still plainer : as yet 
they refer only to the quantitative relations of the combining 
elements, and to some extent to the supposed form of their 
union what letters are to stand for the several elements, 
and how their sequence is to denote the arrangement of 
those elements, we must of course learn or know by heart, 
as both can only be determined by convention : but no one 
can tell merely by looking at the formula thus constructed 
whether it stands for a gas or a fluid or a solid body, nor 
what its density is or its specific gravity, nor what its colour 
may be, whether it is fixed or volatile, soluble in water or 
insoluble. If a man after looking at the formula answers 
these questions correctly he does so upon the basis of 
analogies with which his experience supplies him, and which 
he could not draw from the formulae themselves with any 
certainty that they would be correct. And yet all that is 
wanted here would be the determination of properties or 
modes of relation, which though not absolutely homo- 
geneous are yet as physical processes dependent upon one 
another and functions of one another, and therefore give 
room for t hope that laws may be discovered which will make 
it easy to mark by signs their dependence upon each other : 
but the difficulties would be vastly increased when we tried 
to characterise all our conceptions and had to deal with the 
combination of unhomogeneous elements which yet have a 
necessary relation to each other. 

198. But it is not a system of signs only that we want, 
nor is the success of mathematics due to its symbols, 
though the skill with which they have been chosen has 
no doubt furthered its advance. The truth is that the 
usefulness of the signs rests here upon the fact that we 
already have unambiguous rules, which enable us to deter- 

T 2 


mine what follows from the simplest combinations of 
magnitudes, and then being applied anew with the same 
freedom from ambiguity to the results thus obtained issue 
in these elegant and certain methods of solving problems. 
It is these rules that we must feel the want of when we 
try to combine conceptions which denote something more 
than magnitudes so as to produce a certain result; and 
I believe that we have absolutely no reason to flatter 
ourselves with the hope that these rules would of them- 
selves suddenly become perfectly clear so soon as we 
had analysed into their ultimate constituents the essences, 
contents, and matter to which they were to be applied. 
Assuredly there is no need to insist on the fact that 
increased clearness in the objects cannot but have a 
favourable effect on the certainty of our conclusions re- 
garding them; but in the main it is not by analysing 
our conceptions and tracing them back to primary con- 
ceptions, but by dissecting our judgments and tracing them 
back to simple principles that we must hope gradually 
to fix our convictions which on so many points are still 
in flux. 

But there are two things which we shall require to know : 
first what are the necessary consequences which follow 
from certain definite relations which, as we either arbi- 
trarily assume or are forced to believe, hold between the 
contents of various conceptions ; and secondly what general 
laws, not proved to be necessary but found to hold good 
in fact, connect various ideas in such a way that our 
reason, founding upon these laws, can deduce the con- 
sequences that will then necessarily follow from given 
conditions. These problems, which concern the application 
of the form of judgment, we must for the present attempt 
to solve without the valuable assistance which that uni- 
versal system of signs would no doubt afford if it were 
once completed. 

Note on the Logical Calculus. 

THE idea of a logical calculus has been often taken 
up and often abandoned : but the Englishman Boole has 
recently made an elaborate and careful attempt to carry 
it out, which is beginning to attract attention in Germany 
as well as in his own country. Though I freely admit 
that the author's ingenuity makes his able work 1 very 
charming, I am unable to convince myself that this calculus 
will help us to solve problems which defy the ordinary 
methods of logic. 

Boole does indeed insist that the result of a calculation 
when completed must be expressible in logical terms ; 
but he holds that between the statement of the problem 
and its solution a course of operations may be introduced 
whose several steps allow of no logical interpretation ; and 
he appeals to the extension of mathematics by the intro- 
duction of imaginary quantities. This appeal is hardly 
relevant. The mathematician could not avoid imaginary 
formulae : he lit upon them in the course of well-founded 
calculations : he has always sought for the interpretation 
of the enigmatic expression and has actually found it in 
the province of geometry. In the logical calculus on the 
contrary this working in the dark to which recourse is had 
from time to time would have to take place by means 
of symbols which have been arbitrarily chosen to denote 
logical elements and the relations of these elements. If 
therefore a calculation is really of use only when it allows 
us to solve single problems mechanically, without requiring 
us to be conscious at every moment of the logical meaning 

1 [' An Investigation of the Laws of Thought/ London, 1854.] 


of what had taken place, it becomes all the more necessary 
that the rules which make such labour-saving processes 
possible should be determined upon purely logical prin- 
ciples without any rash and misty analogy from the province 
of mathematics. Though on this point I entirely agree 
with the admirable exposition of Schroder 1 , yet I cannot 
entirely follow him : his demonstrations, which after the 
manner employed by mathematicians follow upon the 
statement of the theorems to be proved, have in my opinion 
no significance beyond that of establishing that the whole 
calculus is consistent with itself and that all the trans- 
formations and combinations of its elements which it allows 
lead to the same results when applied to the same pro- 
blems : but we can only feel confident that the calculus 
as a whole is applicable, when it has been directly shown 
that each universal proposition is only the transcription 
of a logical truth into the symbolic language that has been 

It has long been the custom in the section of logic that 
deals with artificial classifications to make use of letters to 
denote the marks which combine in various ways to form 
the different species that fall under a concept. Supposing 
that the three marks ABC belonged to the general notion 
My the principle of disjunction would direct us to reduce 
each of them to its subdivisions a^ # ? # 3 ..., h l b^ <f> a ... ; the 
complete set of triplets of the form a b c, of course not 
counting repetitions or permutations, would represent all 
the kinds of J/, which, failing any closer determinations, 
may be regarded as equally possible. These groups ob- 
tained by combination express per se merely the simul- 
taneous presence of their elements ; they leave the nature 
of the connexion between the latter undetermined in two 

P'irst of all they do not assign the final form which is to 
be the result of the completed combination. Where logical 
1 [' Der Operationskreis des Logikcaldals,' Leipzig, 1877.] 


classification is aimed at this want is supplied by the image 
which is retained in thought of the abstract M, of which the 
kinds are in question ; this M is to be added in imagination 
to each combination a b c, as the general outline which the 
union of the elements is to fill in ; apart from such an 
occasion for the procedure by combinations, a b c taken by 
itself only designates any object of thought, no matter how 
constituted, in which the marks , b, and c are found to- 
gether, or what is more important, any case, which it has 
not yet been possible to characterise more closely, in which 
the conditions a, />, and c are found together. This uncer- 
tainty does not exist in mathematics, for the form which the 
result of the calculation is finally to take, is here completely 
and solely determined by the definitely assignable nature of 
the connexion which this science requires to be introduced 
between its elements. 

Now with regard to this second point also, the reciprocal 
determination of their component parts, the formulae em- 
ployed in the combinations, in themselves, contain no 
explanation of any kind. In algebra custom has made 
them an expression of multiplication ; the particular sign of 
this operation which has to be retained in the case of arith- 
metic has been found unnecessary, in the case of algebraic 
calculations at least, and the product of multinomials has 
been found equal to the sum of the combinations of their 
elements. Logic, on the other hand, does indeed pre- 
suppose every mark that belongs to a whole to be connected 
in a particular way with every other, but it has no means of 
actually expressing these specific determinations, and en- 
trusts them to our independent knowledge of the subject. 
But what universal laws it does possess on its own account 
with regard to the connexion of the marks bear no re- 
semblance to the idea of multiplication. I will not here 
lay much stress on the fact that the multiplier, which must 
be thought of to begin with as a whole number, leaves the 
value of the multiplicand as a separate number unaffected 


and only repeats it several times over ; while every mark <:, 
which is annexed to a combination a b> not only modifies 
the reciprocal determination of these original elements, but 
at the same time by adding to the matter of the thought 
limits the extent of its application. Anyone who cared to 
dispute the question might perhaps find it easy even on this 
point to make more of the analogies between the two sets of 
relations than of their differences. But it is an essential 
fact for our purpose that while multiplication is forced to 
retain both the recurrences a a, b b> and the permutations 
a b, b , as indispensable components of its product, logic 
can admit no meaning in the former and no distinction 
between the latter. Thus the nature of the case presented 
no occasion for departing from the neutral significance of 
combination-formulae which can have many kinds of mean- 
ings, and applying to them the mechanism of calculation, 
which has strictly speaking no suitability to them except as 
symbols of quantities that can be multiplied. It could only 
be ventured on in the hope that the more extended applica- 
tion of the calculus would compensate, by results which no 
other means could attain, for a cumbrousness inevitable at 
the outset, seeing that exceptional rules were necessary to 
bring such an inappropriate mode of calculation into har- 
mony with its logical object-matter. 

Every A, according to the law of Identity, must = A. 
Natural thought has no motive to determine such an A 
over again by a characteristic A, in the same way in which 
A would be determined by a second mark b. No doubt we 
speak of a human being as truly human, or emphatically of 
a man who is indeed a man ; but we only employ such 
expressions where it is permissible to distinguish the con- 
ception M of an ideal from the conception /jt of the particular 
facts from which the realisation of the ideal is expected. 
At bottom, therefore, we are not determining a single M by 
itself. The human being M /ut that is thus pronounced to 
be truly human, corresponds to its determination M once 


only and then completely, and just so in another aspect 
corresponds to its zoological conception ^ once only and 
then completely; such a thought bears no resemblance to 
the attempt to determine quadrupeds over again by repeat- 
ing the character ' quadruped.' Nothing but the machinery 
of the calculus can suggest the requirement that a should be 
determined by a as in multiplication ; but then the formula 
a a = a or # 2 a which is now introduced to restore logical 
truth, should at least abstain from professing to be a newly 
discovered fundamental law of thought, or indeed anything 
but a make-shift contrivance to correct an improper proce- 
dure. The determination of a by a is logically speaking an 
operation that cannot be performed ; it is only because and 
in as far as, in the context of our thoughts, such a fruitless 
attempt does not result in cancelling the a on which it is 
made, that it is permissible to substitute a by itself for the 
a' 2 to which the calculus would bring us ; but by no means 
to treat this a 2 as existent, and pronounce it equal to a. 
The left side of this equation contains an insoluble problem; 
the right contains, not the solution, but what has to be ac- 
quiesced in because there is no solution. 

This is no mere verbal dispute, as may be seen from 
some considerations which Boole subjoins. If we accept 
a" = a for an equation, it is an easy step to the inferences 
a' 2 a = o or a a* = o; Boole resolves this last formula 
into a(ia) = o. Now the law of excluded middle 
teaches us that everything that is thinkable is either a or 
not a ; this truth is expressed by Boole, who indicates the 
totality of the thinkable by the symbol i, by saying that 
not-tf is what remains of this totality when we subtract a 
from it; so that (i a] is the contradictory opposite of a. 
Now the meaning of giving the equation the form in which 
one side is zero can only be that the combination on its 
left side has no extensi6n that falls under it, and cannot 
therefore occur at all. Thus the formula a(ia) = o 
becomes the expression of the law that nothing thinkable 


can be at once a and not-a. We may be delighted with the 
plasticity of the calculus which furnishes such a graphic 
expression of a familiar truth ; but we shall be the less 
prepared to admit the interpretation which Boole gives his 
formula on p. 50 of his work. It shows, he contends, that 
the law which is regarded as the highest principle of meta- 
physics is only a consequence of a law of thought which is 
really mathematical in form ; that it is because this law 
finds expression in a quadratic equation that our divisions 
and classifications have to be performed by dichotomy ; and 
that if the equation had been of the third order we should 
have been forced to proceed by trichotomy. 

I am sure that I shall not be guilty of trichotomy in the 
sense of hair-splitting if I object to this extraordinary piece 
of argument. Boole himself mentions that from a 2 = a we 
can further deduce a' 6 = a, but he disposes of this cubic 
equation with the remark that two of the factors which it 
presupposes, (i -f- x\ are incapable of logical signifi- 
cance ; and it was clearly the same reason that decided him 
at an earlier stage to attach his inferences not to a' 2 ~a = o 
but to a a 2 o. This procedure implies an idea which 
is quite correct; among the numerous formulae which 
can be mathematically derived from the supposed logical 
principle a* = a none have any meaning but those which 
express something that is of use in logic ; the validity of the 
logical law does not depend on the shape of the formula ; 
it is the value of the formula as a symbol that depends on 
its agreement with the import of the law. But the quadratic 
form itself and its interpretation are altogether a mere 
caprice. I shall not insist on the point that according to 
a 2 = a, a should have been at once substituted for # 2 , 
which would have brought us back quite intelligibly to 
a a o', for even if we believed it possible to retain a 2 as 
a real result of a practicable determination of a by a, and as 
such to equate it with a, still there was no sort of logical 
justification for resolving a a 2 into a (i a). In mathe- 


matics, where we are speaking of magnitudes, the trans- 
formation is correct and in it i really means unity ; but in 
logic the difference a # 2 does not present the least 
motive for regarding it as the product of two factors. The 
i, which is introduced in doing so, is not unity, which it 
would have to be if the resolution were to be mathematically 
correct, but is Boole's arbitrary though not inappropriate 
symbol for the totality of the thinkable ; the truth that a and 
i a taken together exhaust this totality must therefore 
be established to begin with, in order to so much as make 
the interpretation possible by help of which the formula is 
intended to yield it. 

These chimeras have not found their way to Germany ; 
but I have mentioned them at length because of their con- 
nexion with a general conception which does meet with 
some assent among us. We do not overlook the differences 
between arithmetical and logical computation ; but there is 
an inclination to the idea of a more general mathematical 
calculus 1 , for which this distinction of object-matter would 
be indifferent. And it is true that every single act of 
thought, apart from the logical import of its result, admits 
of many uniform repetitions, and the result admits of many 
connections and rearrangements ; further, the notions of 
equality, inequality, and opposition have significance even 
where tfyey do not relate to magnitudes ; though what 
consequences they have in such cases must of course be 
determined for each sphere according to its peculiar nature. 
Still, when it has been determined, when, that is, it has 
been decided under the jurisdiction of logic, what result 
must be derived from the combined or separate occurrence 
of several acts of thought and their particular results ; then 
the recurrences and inter-connexions of all these elements 
may be embraced under the same rules of union, severance, 
and arrangement which hold good of all that is recurrent 
and that has number. Only the laws which are specifically 

1 [' Eines noch allgememeren mathematischen Algorithmic ' ] 


logical and, like the law of excluded middle, govern the 
formation of the actual elements which are to enter into this 
new connexion, must stand on their own feet ; and it is an 
idea as incorrect as it is confused to expect that they can be 
established by any mathematics however abstract which 
should still merit that name in contradistinction to Logic. 
On the contrary, all that such a science would have to teach 
would be the development of the simplest logical truths, 
which are uniformly true of the manifold and its combina- 
tions, whether those of what has number and is homogeneous, 
or those of what has mere relations and is heterogeneous. 
Many things may be proved by mere verbal deductions ; 
and so it may be held an important task to reckon up these 
truths, in their abstract form apart from their applications ; 
I think it rather tedious than indispensable. 

As direct expressions of such extremely simple truth we 
at once think of the axioms, the separate introduction of 
which is hardly more than a matter of form. Obviously the 
logical calculus must agree that a a, and that every a and 
b which are equal to a third thing c are equal to each other ; 
only the definition of equality demands a few words. Logic 
uses a to indicate a general mark, a general class, or a 
general case ; and is therefore able to accept the language 
of the calculus, according to which a is the symbol of a 
class, whose extent comprehends all individual things or 
cases of whatever nature which share the character a. 
These relations of extent are all that the calculus notices ; 
it therefore sets down two class-symbols, a and />, for equal 
when they present to thought classes composed of identically 
the same individuals and are therefore only two names for 
the same class. In such a case a and b may be different in 
themselves, even if their extensions are fully coincident ; 
thus equilateral and equiangular triangles, if nothing but 
their extension is considered, are of course merely two 
names for the same class ; still in logic we could not pro- 
nounce the two conceptions equal as regards the contents 


which they directly declare as their own meaning. It 
follows just as simply from those simplest truths that it is 
always possible to comprehend two acts of thought and 
their results in a sum a + b\ that a b is also possible in 
logic if the necessary homogeneity is obtained by b being 
included in a ; that the other combination a b, which col- 
lects the two characters into one idea, represents a new 
class-symbol with a defined extension ; and finally, that 
where the problem put before us is only that of carrying 
out some uniform mode of connexion, no difference can be 
made by the order of the summanda or factors which we 
combine to make a sum or product. 

These easy analogies between mathematical and logical 
reckoning are less deserving of mention than the differences 
which are derived from the specific nature of logical thought. 
I have already mentioned the equation a 2 = a and not less 
paradoxical is the form in which the law a-\-a = a veils the 
logical truth that each universal conception exists once 
only, that therefore every logical assertion about what 
comes under such a notion is completely exhausted when 
it is once thoroughly admitted of the conception itself, and 
that no new truth can be obtained by repeating the same 
process on the same object. Just so the theorems a + al> = a 
and a (a + b) = a remind us that every assertion which is 
once granted to be universally true of a is also true of every 
species of a that is still further determined by any mark , 
and that therefore the mention of ab beside a remains in- 
effectual, in other words, the former is ' absorbed ' by the 
latter. It is only the improper employment of the sign of 
equation that gives these theorems their appearance of 
peculiarity; all that they really say comes to this ; wherever 
the mechanism of the calculus would naturally lead to the 
forms # 2 , a + a, a + ab, these useless incidents of its method 
are to be replaced for logical purposes by a simple a. 

More important is the extended use which the calculus 
makes of the law of excluded middle ; for the principle of 



Duality, which appears at this point as a new law of 
thought, conceals nothing more than this familiar law. - If 
we use a' to designate the contradictory opposite of a, and 
i for the totality of the thinkable, then we have, really as 
equations, the formulae a-\-a' = i, according to which all 
possible matter of thought is exhausted by a and not-#, and 
a a' = o which declares the impossibility of a union of a and 
not-tf. No further proof is either possible or necessary, 
whether for these laws or for the remaining one that the 
negation of not-a brings us back simply to a and not to any 
third thing ; they are logical truths which have no doubt 
received in those formulae a very clear and convenient 

The old Logic had its chapters about immediate infer- 
ence, conversion, and contraposition of judgments, and 
endeavoured by help of this same law to pursue the content 
of an enunciated judgment into its relations to judgments 
not yet uttered. Boole in a more comprehensive spirit sets 
before himself the problem of developing the different 
and mutually exclusive divisions of the thinkable that may 
be formed by the affirmation and denial of the concepts, 
class-symbols, or elements of whatever kind united in 
a judgment. If x and y are the given elements, and x* and 
y their contradictory opposites, then xy, xy', x'y and x' y f 
are evidently the four classes into which all that is thinkable 
must be divided ; that is, the constituent parts of the com- 
plete division which Boole calls the expansion or develop- 
ment of the given relation between x and y. It is some- 
what inconvenient that following mathematical tradition 
he designates that relation between x andj as a c Function ' 
of the two,/(#, j/); logically such an expression can mean 
nothing, unless it is understood as the definition or predicate 
of some M, for then all the constituents xy, xy\ etc., 
might be deduced from the given connexion between x and 
y, and with them the coefficients which would indicate them 
as possible or impossible within the extension of M. Boole 


however employs for the moment the independent function 
f(x,y] in order to develop out of it in an equally general 
way the law of the formation of those coefficients. His 
original equation x 2 = x, as he can find for it only the 
two arithmetical analogies o 2 = o and i 2 = i, induces him 
to make the assumption that the logical and the mathe- 
matical calculus would completely coincide if these two 
values were the only ones which any magnitude could 
assume ; and conversely, he takes all mathematical opera- 
tions to be permissible in logic, on condition that the 
class-symbols to which they are applied are treated as 
magnitudes which admit of these two values only. So 
taking ax-}- bx' as the given function /(jc), and/(i) and/(o) 
as the two values which it assumes if we take x = i and 
x = o (x' always assuming the opposite values), it is shown 
that/(^) may be obtained by the combination of the two 
values : f (x) =/(i) x H-/(o) x. The same consideration 
leads, in the case in which the given function contains 
the two elements x and j/, to the formula : 
f(x,y) =/(i, i) *y +/(i, o)*/+/(o, i)*>+/(o, o) #y, 
in which the two bracketed values refer in their order to 
x and y respectively. 

If any stress is to be laid on this scheme of the logical 
development of a function, it would have been easy to 
establish , it in a less bizarre fashion. It must after all 
be borne in mind that the zero which denies every magni- 
tude alike, so that for every ;;/, o . m = o invariably, and the 
unit which every magnitude contains as a silent factor, 
so that for every ;//, i . m = m invariably, are exceptional 
and not merely homogeneous with all other magnitudes 
even in arithmetic. Granted that they rank as magnitudes 
when considered by themselves, still in combination or 
multiplication with other magnitudes they have the general 
logical import of affirmation and negation. What was 
required in the above theorem was only this logical mean- 


ing, valid indeed for arithmetic but not derived from it; 
it was therefore improper to give currency to the illusion 
that logic is indebted to the peculiar laws of arithmetic 
for the instruments with which it operates. I will take two 
examples to show what I mean. 

First, if M = a x + 1 x' , the value of the right side will 
obviously be reproduced if we first suppress the first term 
and leave the second, then suppress the second term and 
leave the first, and finally add together the two that are 

a x 4- b x' a x 4- o . b x' -f b x' + o . a x ; 

then the coefficients can of course be expressed by/(i) and 
/(o), and 

Again, let the function f(x, y}~ax-{-by be given and its 
development with reference to the terms xy, xy' , x'y and 
x f y r required ; and further, to make sure of what we are 
speaking, let us regard/ (x,y) at the same time as a definite 
M, whose definition, or specification of extent, is contained 
in the right side of the equation. 

Within this J/, the combination xy is possible in three 
cases, being the a #'s which are also y, the by's which 
are also ^, and the a x's which are also by in full, or 
the ^ys which are also a x in full ; for none of these 
combinations are expressly excluded by the right side of the 
equation. We should therefore get a x y, bxy, abxy; 
but as logically speaking the a b are included besides both 
under a and under #, it is sufficient to exhibit a + b as 
coefficients of xy\ apd of course a-f ^=/(i, i) is equal, 
that is, to the value of the right side for x i,y = i. The 
second term of the development would contain xy' } the 
equation tells us that if we suppress by which can never 
be combined withj/ there can occur within the compass of 
M no y or not^v besides a x ; consequently a is the co- 
efficient of xy f , and a of course =/(i, o). Just in the 


same way it follows that within M there can be no other 
x f or not-# but by \ consequently b x f y is the third term, 
and b of course =/(o, i). Finally the equation tells us that 
the extent of M is entirely exhausted by a x and by^ and 
contains nothing that is neither x nor y ; hence o is the 
coefficient of x* y f , and it again =y (o, o). 

Thus there is no doubt that the proposed formula of 
function-development can be justified from purely logical 
considerations, and I would attempt to establish this on more 
general grounds if I saw more clearly what is the purpose 
of the whole proceeding. The first examples which Boole 
gives can only be regarded as exercises. If clean beasts x 
are according to the Jewish law those which divide the 
hoof y and chew the cud #, and then the development tells 
us there are no clean beasts which divide the hoof but do 
not chew the cud, and none which chew the cud but do 
not divide the hoof ; that again there are no cleitn beasts 
which do neither the one nor the other, and lastly there can 
be no beasts which do both and yet arc not clean ; I have 
my doubts of the frequency of the logical desire to go 
through these deductions of given fact ; but if any one feels 
the want, it is beyond a doubt more easily satisfied without 
a calculus than with one. But there are two other problems 
which Boole hopes to solve by help of such use of formulae ; 
first, if a number of elements are given in any combination, 
the equation which expresses this combination is to be 
solved with reference to any of its elements at pleasure ; 
and then it is to be possible to eliminate any one from the 
equation, in order to display the relations of the rest to one 

As regards the first problem, I can only regret that Boole 
abandons himself recklessly to his principle of permitting 
himself all operations of reckoning if only their result can be 
logically interpreted. From the given proposition 'All menj> 
are mortal xj he obtains by contraposition * No man is not- 
mortal' yx'=o. Now as x' + x = i arid therefore #'= i .r, 



we get y (ix) = o or yxy~o, xy~y } then further 
#=- and by development of- we obtain x=y+- (i~j) 

or =y + -X; this he takes to mean, introducing the mathe- 

matical significance of the symbol - : ' mortal includes all 

men and an indefinite number of what is not man.' Results 
that could only be obtained in such unwarrantable ways 
would certainly form no extension of Logic. Moreover, in 
this case such arts were not even necessary. For not the 
contraposited form 7(1 #) = o but the original y x 
should have been employed, only with the precaution of 
providing x from the beginning with a particularising factor 
77, y = v x. The proposition 'All men are mortal' means 
simply this and nothing more in the world besides ; it merely 
regards y as subordinate to an x within the compass of 
which there is something else as well. There is no possible 
meaning in finding over again by calculation precisely what 
was presupposed, and what is self-evident, that is, that x 
comprehends beside the v x which are y a further in- 
definite number of kinds which are not y \ that therefore 

With respect to the process of elimination, I shall con- 
tent myself with giving an example. Every logical equation 
can, by applying contraposition to the affirmative judgment 
which it expresses, be reduced to the form in which one 
side is zero ; for the equation x z = o simply means that no 
x is z. I pass over Boole's doctrine about the procedure of 
collecting ail given single judgments or equations into one 
solitary resultant equation, and suppress the scruples which I 
feel as to the necessity or productiveness of such an operation. 

It is granted then that the equation is to be presented in 
the following arrangement; pab + qab' + ra! b+sa!b r = o ; 
then the product of the coefficients pqrs equated to zero, 


is assigned as the result of the simultaneous elimination of 
# and b. This is easily seen with the ordinary appliances 
of Logic. For logically this equation cannot have the value 
o unless each of its terms taken by itself o. Further, 
pab o says that No/ a is b ; but qab f o gives by contra- 
position All g a are b, and so in Cesare, No q a is / a, or, 
_pga o, and this again gives No/f is 0, or by contraposi- 
tion, All / q are a'. Again r a' b = o gives, No r a' is b ; 
but .$ a' b'~ o gives by contraposition All ^ a' are b ; so we 
get in Cesare, No s a' is ra', or, s ra f o, or, No r j is a'. 
If we subsume the second conclusion No r s is a under the 
first All p q are a\ there follows in the same figure, No r s 
is / q ov pgrso. It is easy to see that if a similarly 
arranged equation with one side zero contains besides a, b, 
and a ', b 1 ', more such pairs of opposites r, /, the elimination 
may be continued in the same way. But no doubt for such 
cases there is value in the abbreviated rule that the result of 
the elimination consists in the equation of the product of 
the coefficients to zero. If the equation had contained 
besides a term z = o independent of the pairs to be elimi- 
nated, it would persist without change, and might be added 
to the preceding term, so that in the result p q rs + z = o 
each of the terms by itself is = o. Schroder remarks on 
this question at p. 23 of his work that the results of the 
elimination of a symbol a from several isolated equations are 
less comprehensive than those of an elimination from the 
combined final equation; xa+ya'=o and / a + q>a' = o 
when taken apart, only give xy o and/ ^ = o; while on 
the other hand the combined equation gives xy+$x+ 
py +p q = o ; and for this reason he thinks the latter order 
of procedure preferable. Is he not in this artificially 
making little difficulties, simply out of the order of procedure, 
which must ultimately depend on the development of the 
functions? Why are we forced to unite the four terms 
x a = o, y a? = o, p a = o, and q a' = o, although they must 
be true by themselves, in two equations, instead of regard- 

U 2 



ing them as four terms to be employed at pleasure ? Then 
we might find without difficulty all results of elimination 
which we had any interest in ascertaining. 

I do not maintain that the same syllogistic process will 
easily bring us to our goal in every case, especially in more 
complicated cases. But Boole himself insists that we must 
carefully analyse what we mean in every case, before trans- 
lating our notions into the language of the symbols ; and 
I certainly believe that the fulfilment of this postulate 
would enable us to dispense altogether with the calculus, 
and that Logic would prove rich enough to allow of the 
invention of adequate means of solution corresponding to 
particular problems, even if these means were not stereo- 
typed beforehand. With reference to this point I mention 
a problem which Boole 1 puts and which Schroder repeats. 

It is assumed to be known from an analysis of experience 
that in a certain class of natural or artificial products the 
combinations of the marks a b c d e are subject to the 
following rules ; and in such a way, that not only the 
occurrence but also the non-occurrence of each particular 
mark belongs to the conditions from which the presence or 
absence of the others has to be inferred. 

1. Wherever a and c are absent at the same time, e is 
present, together with either b or d, but not with both ; 

2. Where a and d occur, but not e, b and c will either 
both be found or both be missing. 

3. Wherever a is found in conjunction with either b or e 
or with both at once, either c or d will be found, but not 
both together. 

4. Conversely, where, of the pair c and d, the one occurs 
without the other, a will be found in conjunction with either 
e or b or with both at once. 

It is required to ascertain : 

i. What can be inferred from the presence of a with 
reference to b, c, and d ; 

1 [' Investigation of Laws of Thought/ p. 146 ff.] 



2. Whether any relations, and if any, what, exist between 
, t, and d, independently of the other marks ; 

3. What follows from the presence of b with respect to 
tf, <:, and d^ and 

4. What follows for a, c y and d independently of the 
other marks. 

Boole anticipates that no logician would find the right 
answers to these questions by syllogistic process, unless he 
knew them beforehand ; I fully admit this, but who would 
be tempted to select that process for attacking this problem 
while the more suitable one offers itself spontaneously? 
We" have only to make a list (it is a purely mechanical 
process) of all the combinations of five which can be formed 
out of a I c d e and a' // c f d' e', avoiding repetitions and 
the inclusion of contradictory elements, and then, or while 
making the list, to suppress those which are excluded by 
the totality of the given conditions. This leaves only TI 
combinations ; 

abcd'e ab' c d' e a' b c d e a'b'cde 
abcd f <f ab'c'de a' b c d e' #' b' c d e f 
abc'de ab'c'd'e' a f b c f d' e 

From these we can read off the answers to the questions 
proposed : 

i J . We infer from the presence of a that either c or d 
is present, but not both, or else that <*, r, and d are all 

2. There is no independent relation between <, c^ and d, 
for all conceivable combinations of them with b', c', d' are 
equally realised. 

3. From the presence of b it follows that either a, t, 
and d are all absent, or some one alone of them is 

4. If a and c are both present or both absent, d is im- 

Similar questions about e which are not proposed could 
1 [Cp. Boole, pp. 148-9.] 


be answered out of the same conspectus without any 
distinct operation. 

I borrow from Schroder's treatise for purposes of com- 
parison no more than the beginning of the solution by 
calculation; not so much to show that if all the inter- 
mediate terms are actually supplied it is by no means 
distinguished by brevity, but chiefly with the general object 
of elucidating the use of the calculus by help of an instance 
that involves a real problem, and is not merely going back 
upon what we know to clothe it in awkward formulae. 

By contraposition of the positive judgments which con- 
stitute the given conditions of the possible combinations, 
and so reducing them, as equations, to the form in which 
one side is zero, we obtain 

from i. a! c f [>'+ bd+Vd*] = o; 

from 2. a d \b c f + b' c\ e'= o ; 

from 3. a \b + e] [cd + c'd'] + [ad'+Sd] [a'+fS] = o. 

As the questions ask nothing about e and e' the first 
operation to perform is the elimination, which we dispensed 
with, of this pair of opposites. According to the rule given 
above its result consists of equating with o the sum obtained 
by adding those components of the equations which are free 
from e and / to the product of the coefficients of e and /. 
Now to begin with, the coefficient of e in 3. = a (c d + c' d'\ 
and that of e' in i. 2. and 3. = a' c' '+ a d\b / -J- b f c] -f 
V [c d f + c' d} -, the product of the two is according to the 
above-mentioned rules = a b' c d, and with the addition of 
the terms free from e and /, which are = a f c' [b d + b' d'] 
f a b \c d + c' d'} + a' [cd'-\- c' d] the entire result of the 
elimination would have to be brought together into 

a \cd + b c'd] -f a' [c d'+ c ' d + b' c' d'} = o. 
Now to answer by this result in the first place the second 
question, about the relations between , c, and d, we should 
have to eliminate a and #'; but the requisite product of 
their coefficient is = o because each individual product as 


it arises takes independently the value o owing to the 

combination of contradictory elements ; the result is there- 
fore = 0, and we must accept this as a sign that there is 
no independent relation between these three marks. How- 
ever, we see at once that if we give the symbol p to the 
coefficient of a that of a f will become/' or not^ ; we there- 
fore obtain from a p -f a f p' == o the two equations ap = o, 
or No a is /, and a' p f o, or No not-0 is not/. The 
first of these gives at once ; all a are not/, or p' \ hence 
a = c d' + c' d + tf c f d', which formula answers the first 

I omit the continuation which would be needed to answer 
the third and fourth questions, and confine myself to re- 
marking that in the whole of this problem no use has been 
made of the development of functions, of the importance 
of which I expressed my doubts above ; the required 
equations were obtained directly from the given proposi- 
tions, and the eliminations out of them were conducted on 
a method, the origin of which we explained to ourselves by 
help of syllogisms in the second figure. Thus there is 
nothing to be said against the appropriateness of the present 
method ; but just as little against the superior simplicity 
and plainness of that which we adopted. This, by the way, 
had not to wait to be discovered by Jevons, for it was 
already fprthcoming in the doctrine of classification, which 
long since required in the first place the tabulation of all 
the marks in their combinations, and then the cancelling 
of all combinations that become inadmissible on taking 
account of the reciprocal determinations of the marks. I 
cannot therefore convince myself of the advantages to be 
derived from the attempt to systematise in a fixed logical 
calculus all the means of vivid and abbreviated presentation 
to which every one has spontaneous recourse in given cases, 
applying them with variations adapted to the proposed 
problem. It is inevitable that a symbolic method intended 
to make uniform provision for every case should purchase 


its suitability for the solution of one problem at the cost of 
a useless prolixity in its treatment of others and of manifold 
discords with the custom of language. 

Even the quantification of the predicate, which was the 
starting-point of recent English logic, was no new discovery, 
but the superfluous inflation of a familiar idea to an ex- 
cessive importance. That the predicate of a judgment, 
except in case of simply convertible judgments, has a larger 
extent than the subject which in part takes its place within 
this extent; that therefore it is not merely the predicate 
that determines the subject, but also the subject that 
restricts the predicate to such a modification as is true of 
the subject's self; these were old doctrines of logic, and in 
its rules of conversion it went so far as to provide for their 
application. It is true that the scheme of judgments gave 
no special expression to this truth, just as the ordinary 
linguistic form of the sentence did not. But what harm 
was there in that, when the fact was known ? Did the want 
of such an expression ever deceive a considerate thinker ? 
And was it worth while, for the sake of amending such 
trifles, to have recourse to such dangerous contrivances, as 
to connect the natural expression of thought with a new 
symbolism and a new calculus ? There could be no real 
gain in expressing the proposition 'All men are mortal' 
by y = v x unless a means could be discovered of defining 
this v ; as long as it remains an undefined coefficient, it is 
an ineffectual indication of what we knew before. In the 
converse of this judgment c Some mortal is man,' the old 
logic would bring to light this indefinite Particularity 1 
neither better nor worse than that v would; if we object 
to the expression 'some,' our objection might be easily 
removed by the consideration that these indefinite particular 
judgments are at the same time forms of modality, and 
express the possibility of a conjunction of their predicate 
with the general notion which forms their subject, by 
1 [' Diese unbestimmte ParticuUiitat.'] 


affirming such a connexion for some but not for all cases 
of the notion. 

There is a passage of Jevons (' Principles of Science,' 
London, 1877, p. 59) which among others has occasioned 
these remarks. He forms two premisses; sodium 1 = sodium 
metal, and sodium = sodium capable of floating on water. 
He draws the conclusion sodium metal = sodium capable 
of floating on water. To this he subjoins these remarks. 
" This is really a syllogism of the mood Darapti in the third 
figure, except that we obtain a conclusion of a more exact 
character than the old syllogism gives. From the premisses 
* Sodium is a metal ' and ' Sodium floats on water ' Aristotle 
would have inferred that 'Some metals float on water.' 
But if enquiry were made what the ' some metals ' are, the 
answer would certainly be 'Sodium 2 .' Hence Aristotle's 
conclusion simply leaves out some of the information 
afforded in the premisses; it even leaves us open to in- 
terpret the c some metals' in a wider sense than we are 
warranted in doing. From these distinct defects of the 
old syllogism the process of substitution is free and the new 
process only incurs the possible objection of being tediously 
minute and accurate." Oh no ! we might admit the ' te- 
diously,' but otherwise Aristotle is in the right. Jevons' 
whole procedure is simply a repetition or at the outside an 
addition^ of his two premisses ; thus it merely adheres to 
the given facts, and such a process has never been taken 
for a Syllogism^ which always means a movement of thought 
that uses what is given for the purpose of advancing beyond 
it. So the combination of words which he proposes is not 
a syllogism at all, and consequently not one in Darapti. 
The meaning of the syllogism, as Aristotle framed it, would 
in this case be that the occurrence of a floating metal 
Sodium proves that the property of being so light is not 
incompatible with the character of metal in general. If he 

1 [See Professor Lotze's Preface to the Logic.] 
3 I* Metal which is Sodium/ Jevons.] 


expressed this by saying c Some metal is capable of floating/ 

he intended of course not to repeat the premisses which 
were known before ; but to enunciate the possibility of a 
general distribution of this property among metals, as a 
supposition whose correctness in fact there is ground for 
testing further, since it is logically not inconceivable. Even 
the expression ' Some metal ' is at bottom quite correct, for 
Sodium certainly is some metal; the expression does not 
enjoin us to think of other metals at the same time with it ; 
it is true that it does not prohibit our doing so, but this 
need not give rise to any error. 

How often have modern enterprises like these proclaimed 
the dawn of a wholly new epoch in logic, and the fall of the 
contemptible system of antiquity! I am convinced that if 
the ancient logic were to be really forgotten for some 
generations and then rediscovered by some fortunate 
thinker, it would be welcomed as a late discovery, after 
long search, of the natural march of thought, in the light of 
which we should find intelligible both the singularities and 
the real though limited relevancy of the forms of logical 
calculus with which we had made shift so far. 


The forms of Proof. 

199. IT was our business in writing of systematic logic 
to enumerate the various forms of judgments and to point 
out the precise mode of union which in each of these forms 
is conceived as subsisting between S and P or as to be 
effected between them : it is the business of applied logic 
to consider what contents S and P can properly be joined 
in one of these forms of union. Various problems which 
we shall not always hold apart fall within this scope. In 
the first place the communication of the thoughts of others 
gives us numerous propositions of the form S is P y whose 
meaning and purport is perfectly plain, but whose validity 
is questionable : then there arises for us the problem of a 
proof "far fat given proposition T. In the second place our 
own observations may lead us to suppose that between two 
ideas S and P there subsists a relation which if it were 
known could be expressed in a judgment of the form S is 
P : then we are called upon to discover the yet unknown 
proposition T which would be the precise expression for 
this supposed relation. 

These two, discovery and proof, differ only in their 
different use of the same materials. The same combina- 
tions of thought by which the truth or probability of a 
proposition J'was first discovered may always be applied, 
when put spmewhat differently, and sometimes even with- 


out any such transformation, to prove the truth or prob- 
ability of a given proposition T. Moreover it is easy to 
see that the reflexion of the discoverer, if it is not to miss 
its aim, needs at every step slight connecting links, re- 
sembling a proof in form : and conversely that a proof will 
never reach its goal without some inventive play of thought. 
On the whole however discovery reaches farther than proof : 
and so I will separate the two problems, though I shall not 
always avoid the natural mixture of the two. Scientific 
investigations lead to both in about equal measure; the 
needs of life more frequently lead to discovery. 

I find reason however again to divide the first part of the 
subject, and to separate the proof of universal propositions 
from the proof of particular or singular propositions. It is 
true that a universal relation can seldom be established 
between and P without the employment of knowledge 
supplied by experience ; but as such knowledge, if it is to 
lead to universal conclusions, must itself have universal 
validity, we may regard it as knowledge which, though 
originally derived from our experiences, is yet, now that we 
have full confidence in its universality, to be counted 
among the proper instruments of thought. The proof of 
particular facts on the other hand, of historical events or of 
the ordinary transactions of life, can never follow from 
universal propositions alone, not even from such universal 
propositions as are themselves derived from experience : it 
presupposes the knowledge of a number of particular 
circumstances, occurring only here and only here united in 
this precise manner. The preliminary process of getting at 
all these conditions, from which the conclusion is to be 
drawn, requires peculiar instruments which we shall con- 
sider presently. The solution of a proposed problem on 
the other hand, even when the result is to be not a universal 
proposition, but the establishment of a single fact, may be 
connected with the proof of universal propositions : under 
the conditions which here do not need to be sought but 

Chap. IV.] SELF-EVIDENCE. 301 

are given, and so far as they are given, the definite proposi- 
tion T which satisfies them all is always to be found by 
employing instruments of thought which are of universal 
application, and these theoretical results are inaccurate and 
need correction in practice only so far as we have failed to 
state #//the conditions which 7* had to satisfy. 

20 O. Every proof is a syllogism, or a chain of syllogisms, 
which completes the premises required for the given propo- 
sition 7] so that they fit into one another in such a way that T 
follows as their necessary consequence. But the validity of 
every conclusion depends upon the validity of its premises : 
these again might be established by fresh proofs, but this 
procedure would go on ad infinitum without any result were 
there not a number of universal propositions which we 
accept as immediate truths, which therefore neither need 
nor are capable of proof, but are themselves the ultimate 
grounds by appeal to which we may decide in every case 
whether a conclusion is correctly or incorrectly drawn from 
its premises. I do not intend as yet to discuss the question 
of the source from which we obtain these immediate truths : 
we are here concerned only with the mark which justifies 
us in classing a proposition T among the axioms^ assent to 
which we believe ourselves entitled to demand from every 
sane person. Now it is conceivable that, just because there 
is no possible proof of an axiom, this mark may in the last 
resort be nothing but the self-evidence , the immediate 
clearness and certainty with which the content of a uni- 
versal proposition thrusts itself upon us as a necessity of 
thought ; and in fact this is what we always come back to 
in the end. 

Experience however abundantly shows that propositions 
which later generations have proved to be false, were as 
self-evident to earlier generations and produced in them as 
strong a conviction as any propositions whatsoever : rela- 
tions which in the limited sphere to which our observation 
is confined are seen to be permanently present or constantly 


recurring, without any contrary experience to disturb us, 
very commonly assume the appearance of necessities of 
thought. There is only one way of distinguishing the 
spurious self-evidence of a prejudice from the genuine self- 
evidence of a true axiom : we must try whether the contra- 
dictory of T the proposition in question is as impossible in 
thought as T itself seems to be necessary. This test will 
often be quite decisive ; we shall often find to our astonish- 
ment that the attempt to join S and P in the opposite way 
to that asserted by the given proposition T leads to no 
inner contradiction in our thought at all. In that case ^is 
no axiom, but either altogether an error, or a truth that 
holds true in some cases only, or a truth which though 
universally true requires to be proved. In the other case, 
when the contradictory proposition non-jT appears as im- 
possible in thought as T appears necessary, we may with 
greater confidence regard T as an immediate axiom ; but 
the test does not even now give perfect security, for it is 
quite possible that the inconceivability of non-7' and the 
apparent necessity of T may both alike rest upon a spurious 
self-evidence. Should these two simultaneous errors be 
made, logic furnishes no short way of detecting them : our 
mistake could only be gradually amended by our becoming 
aware of the contradictions which experience offers to the 
assumed validity of T, and by a slow and far-reaching 
modification of our system of thought suggested by those 

Such a double error will seldom be found in the case of 
purely theoretical principles, more often in the case of the 
principles upon which our moral judgments are based, and 
which may be classed as genuine or spurious axioms, 
although strictly speaking they do not seem to be necessities 
of thought but only unquestionable truths, and their oppo- 
sites do not seem to be unthinkable but only absurd. 
That you ought to hurt your enemies was for a long time 
generally accepted by the ancients as an unquestionable 


maxim, and the opposite of it regarded as absurd : such 
errors can generally be removed only by a gradual alteration 
in men's habitual feelings. 

201. Supposing now that T is a universal proposition 
whose validity is not axiomatic, i.e. that it is such as to need 
proof, we yet shall not set about proving it till we know 
that T is worth proving. In three cases it will not be worth 
proving. In the first place it will not be worth proving 
if its content is an incomplete, and therefore an indefinite 
thought. A man of untrained intellect, so long as he 
confines himself to the objects which naturally come within 
his scope, is usually conscientious in enumerating and 
examining all the points which are important for the under- 
standing of a fact : he follows the old rule which tells us to 
ask 'quis? quid? ubi ? quibus auxiliis? cur? quomodo? 
quando?' and to omit none of all these questions. But he is 
quite helpless when he wanders off into general considera- 
tions which belong to the province of speculation : he then 
usually does not get beyond a clumsy expression for 
something which he perhaps rightly believes, demands, or 
assumes, but is unable to connect with any determinate or 
determinable points. The philosopher on the other hand, 
revelling in his abstractions, docs not always come to meet 
him half-way, but often contents himself with employing 
conceptions which when severed from their proper applica- 
tion become utterly meaningless : the result is that vague 
theses are nowhere so common as in the attempts of a man 
who has had no logical training to philosophise by the light 
of nature. That God and the world are one is a proposition 
that no one can prove except him who propounds it ; so far 
as his proof is correct at all it is the proof itself that tells us 
what he meant by the proposition : any other person than 
he who propounded it will, if he be wise, attempt neither to 
prove it nor to refute it ; for that God and the world are in 
some sense two is asserted by the proposition itself, for 
otherwise it could not have distinguished them; but that 


they are also one in some one of the many senses of unity, 
may be supposed without more ado. 

That things are appearances is an equally ambiguous 
proposition : the things which appear to our senses are so 
of course, for otherwise they could not appear to us : but 
that the things which though themselves inaccessible to 
observation we suppose to underlie our sensuous perception 
are also appearances is an incomplete thought till we deter- 
mine what is to appear and to whom it is to appear. All 
these and other similar propositions are not worth proof or 
refutation, but are simply to be returned as they are to him 
who brought them, just as in a court of law we refuse to listen 
to a man who complains that he has suffered wrong without 
saying what has been done to him and who has done it. 

202. The second case is when though a perfectly clear 
nominal definition may be given of S the subject, or P the 
predicate of the proposition T 9 the definition contains a 
combination of ideas which can be shown to be impossible, 
or cannot be shown to be real. No one would trouble 
himself to prove or to refute a proposition the subject of 
which is a wooden iron : no one would seriously enquire 
whether this wooden iron will burn in the fire like wood or 
melt in it like iron. There is no such logical contradiction 
in the ideas of ghosts and will-o'-the-wisps, but we defer 
asking whether the former need sleep, and whether the 
latter are attracted by buried metal, till their existence is 
proved. What we here require may be called in general 
the justification of a conception, which must without fail be 
added to its nominal definition when use is to be made of it. 

This may be effected in various ways. If M stands for 
something which is supposed to have external existence, the 
shortest way to justify M is to point at once to an instance 
of it or to a fact in which its existence is given and 
accessible to observation. If M denotes a combination 
of ideas the validity of which means that it can be carried 
out and that its result can be imagined or realised in a 


mental picture, this very realisation of the content which M 
demands, or in other words its construction will justify M 
itself: thus geometry establishes the admissibility of the con- 
ceptions it has formed by presenting in a visible form what 
they till then only contained as a problem, thereby proving 
most conclusively that the problem was soluble. If we can 
neither point out any instance of M nor carry out its con- 
struction, we must at least show cause or give a ' deduction' 
which explains how in connexion with some demonstrable 
reality or in pursuit of some problem we have been properly 
and justly led to form this conception. Such a ' deduction ' 
cannot always directly prove the validity of M in the shape 
in which the conception is presented, but it may always 
show that M is a preliminary designation for some content 
which we are reasonably and rightly looking for ; it remains 
for the further enquiry whose beginning is hereby justified 
to determine whether M itself can be justified as a valid 
conception, or else how its content must be modified in 
order to make it valid. 

The ancients regarded the doubling of the cube as a 
serious problem : but though they could not geometrically 
construct the required line, whose cube should be double 
of a given cube, yet it was all along certain that the problem 
was soluble and that the required line was a magnitude 
which coyld in some way be found. For it could be shown 
that as the side of a cube increases its volume must also 
continuously increase without any alteration in its shape : 
among this infinite series of larger and larger cubes then 
must be found that particular one which is double of a given 
cube, and this implies that its side actually occurs in the 
series of existing lines. We here show cause for the 
necessary validity of that which is sought instead of actually 
realising it in a construction. 

Again it may be doubted whether one and the same con- 
ception of length fits both curved and straight lines; but 
setting this doubt aside it was not unreasonable as things 



then were to hope to find by a simple geometrical con- 
struction^the straight line which is equal to the circumference 
of a circle of given radius ; for it was certain that the length 
in question depends upon the length of this radius and upon 
nothing else. This hope was only banished by the com- 
pletion of the enquiry, which showed that the circumference 
cannot be expressed as a determinate real and algebraical 
function of the radius. In the natural sciences a hypothesis 
often assumes facts which we can never hope to establish by 
direct observation : often indeed we must leave it to God 
and the future to show even the possibility and constructi- 
bility of that which we are for the present absolutely obliged 
to assume. The only way of justifying ourselves in such a 
case is to show from the given facts the pressing need of 
the idea which we employ, with the reservation of course 
that we may at a future time so alter it as to enable us to 
construct it without impairing its usefulness. We shall 
return to this point on another occasion ; for the present it 
is enough to refer to the instances above employed as 
showing what kind of justification is needed for conceptions 
if their union in a proposition is to deserve proof or 

203. We will now suppose that the conceptions which 
are joined in the universal proposition J'have the requisite 
definiteness and validity: but even so we do not start in 
search of a proof that shall exhibit J'as the necessary con- 
sequences of premises that must be discovered, until we 
have got some preliminary warrant that the proposition is 
true as a matter of fact ; for it would be lost labour to 
try to prove what is not even true. If T is a universal 
proposition of whose field it is not easy to take a com- 
prehensive survey, we first try whether T holds good in 
some examples that lie near at hand : a single case in which 
it did not hold good would do away with the universal 
validity of T, and the problem would then be changed 
into finding the conditions under which T has at least a 


partial validity; if on the other hand that which T asserts is 
found to hold good in all the cases of its application which 
we compare, this trial, though being necessarily incomplete 
it cannot prove that T is universally valid, may yet corro- 
borate what it alleges so strongly that it will be worth while 
to search for a proof. This very needful preliminary pro- 
cedure, which will further on take its place among the 
forms of proof, is in fact neglected but seldom, and that 
mostly in cases where the validity of T 7 cannot be tested by 
mere reflexion upon instances supplied by the memory, 
but only by observation or experiment. The courtiers of 
Louis XIII exhausted themselves in ingenious proofs of the 
proposition that a living fish thrown into a bowl full of 
water makes it overflow while a dead one does not, until 
the gardener who was called in made the experiment and 
showed the assertion to be entirely false ; but others make 
the same mistake, and in the less exact departments of 
natural science we frequently find subtle demonstrations 
and explanations of phenomena whose actual occurrence is 
entirely problematical. 

204. Supposing now that this preliminary question is 
settled, and that T is recognised as a universal proposition 
that deserves proof, its truth or falsehood may be established 
either in a direct or in a roundabout way, and this makes 
the first division of proofs. 

A proof is direct when it shows immediately that the 
given proposition T is necessary or impossible ; it is indirect 
(or apagogic) when it establishes the truth or the falsity of T 
mediately by showing the falsity or the truth of its con- 
tradictory non-7! In each case there are two directions 
which the train of thought may take. We may call a proof 
straightforward or progressive when it starts with that which 
in the nature of the thing conditions something else and 
makes that which is conditioned issue from it as its conse- 
quence ; it is a backward or a retrogressive proof when it 
starts from that which in the nature of the thing is con- 

x 2 



ditioned in order to arrive at knowledge of that which 
conditions it. The first form of proof, since it proceeds 
a prindpio ad prindpiatum^ may equally well be called 
deductive, though the opposite name inductive will not 
be found so generally suitable for proofs of the second 
form which proceed a prindpiato ad prindpium. And 
finally there is yet another distinction applicable to both 
these lines of proof : you may go forward (progressively) 
either from general truths to T or from T to its proper 
consequences, and similarly you may go backward (retro- 
gressively) either from T's consequences to 7", or from T 
itself to the truths upon which it is founded. We cannot 
pronounce upon the comparative value of the eight different 
forms thus obtained till we can consider each in reference 
to the problems for which it is usually employed. The 
following survey may enable us to do this. 

205. The first form of proof, which is direct and pro- 
gressive, proceeds from a universal truth, which is placed as 
major premise at the head of the whole procedure ; in the 
minor premise (or in a series of epi-syllogisms, if the proof 
can only be completed in a chain of reasoning) it is then 
shown in what relation S and P which are joined in the 
given proposition T stand to that major premise; and 
lastly the conclusion infers that by reason of these relations 
of S and P the proposition T which was to be proved must 
hold good. If the problem be stated in this general way it 
seems as if all the three figures of Aristotle might be em- 
ployed in this form of proof : the fact is however that the 
first figure alone answers to the spirit of it. I do not reject 
the third figure on the ground that as usually described it 
only gives particular conclusions, while we here wish to 
prove universal propositions ; if we put the particular con- 
clusion 'some are P 9 into a modal form, 'that which is 
*$* may be P,' we get a universal proposition which it may be 
worth while to prove. For instance if we want to produce 
an effect P, and have nothing to get it out of except an 


unpromising material S, we shall be glad to see it shown by 
a syllogism in Bramantip that S and P are compatible with 
one another in the case of a subject M, and that therefore 
S does not always make the desired effect P impossible. 
But the third figure does not exhibit this proof in the 
progressive form. It only states in the premises an instance 
of the coexistence of S and JP 9 from which we may argue 
regressively, ab esse ad posse, to their compatibility, The 
second figure admits universal conclusions indeed, but only 
negative ones ; these too may be valuable, but they cannot 
be obtained by this figure without premises of opposite 
quality, and therefore fail to satisfy us. For a universal 
negative proposition 7J which simply denies a predicate P 
of a subject S because S and P stand in opposite relations 
to a third M, appeals to a mark which shows that S and P 
cannot be combined, but not to a reason which explains why 
they cannot : it merely expresses a fact which is indeed true, 
but remains unintelligible till we have learned in an affirma- 
tive proposition what S really is, and thus now can see that 
because it is this it cannot be the other, viz. P. And so 
the second figure, since it establishes its conclusions by 
proofs which, though appropriate and convincing, give no 
explanation, is also rather regressive than progressive in 
character. And therefore under the head of direct pro- 
gressive .proofs attention has usually been directed to the 
first figure, especially to its affirmative moods, and for the 
present purpose to Barbara exclusively : it is only here that 
we find the subordination of a given idea under a general 
truth, which enables us to understand not only that T holds 
.good, but why it holds good. 

200. This opinion is as old as Aristotle : it is worth 
while to observe however that this form of proof is to be 
regarded as an ideal in another sense than this : it cannot 
fairly claim the praise bestowed upon it unless we succeed 
in filling it with the content which its articulation requires, 
i.e. unless we set down for major premise a general proposi- 


tion under which the special case of the minor premise 
demands to be placed in virtue of its very nature, and which 
theiefore would actually be the reason upon which the 
validity of the proposition to be proved depends not merely 
for our reflexion but in the nature of things. But it is clear 
that we may use the form of this proof without in the least 
satisfying this last condition. Many instances occur, and 
that precisely in the field of mathematics where exact treat- 
ment is required, of propositions that admit of various 
equally convincing proofs all couched in this form of sub- 
sumption, none of which therefore can claim exclusively to 
express the proper connexion and development of the thing 
itself. The possibility of presenting the same idea in very 
various forms without altering its value enables us here to 
subsume it under a great variety of universal major premises, 
and to proceed from any one of these arbitrarily chosen 
starting-points to the same assertion T. I am anxious not 
to be misunderstood here and will therefore go into detail. 

I will in the first place allow that we often find in mathe- 
matics a proposition T which is so evidently only an appli- 
cation of a definite major premise M that its deduction 
from this major alone seems natural, from any other 
artificial. I will remark in the second place that when T 
may be deduced with equal ease from a variety of majors 
M NO y I do not find in this alone any reason for saying 
that these various proofs are foreign to the natural sequence ; 
for it may be (though I do not propound this as the true 
theory but only suggest it as a possible view) that the whole 
of our knowledge (e.g. of geometry) rests in fact upon a 
number of original and equally self-evident perceptions, 
none of which can be deduced from any other, but which, 
like the several components of one complete thought, are 
each and all valid at once and connected in definite ways 
with one another. We could then understand how in 
virtue of this connexion the same proposition admits of a 
variety of equally convincing proofs, according as we start 


from one or the other of those inseparably united percep- 
tions : no one of these proofs will exclusively exhibit the 
nature of the thing, but yet each may actually exhibit it in 
the form in which it is seen from that particular point of 
view ; the possibility of a variety of proofs rests in this case 
upon the organisation of the content itself, which makes a 
harmoniously articulated whole not on one line only but on 
several lines at once. But I must nevertheless add in the 
third place that there remain many propositions 7] whose 
proof (I mean in this form of subsumption) can only be 
effected by devices, which can be justified after they have 
been applied, but to the application of which we cannot 
find any invitation in the thing in question. It is to these 
proofs, of which many occur in pure mathematics, and a far 
greater number in applied mathematics, that the remark 
above made is intended to apply ; though these proofs are 
as conclusive as can be wished, it is yet quite beyond our 
power to take them all in at one view, especially when they 
form chains of many links ; and as they scarcely allow us to 
do more than see the necessary consequence of coupling 
each link to the one which follows, while the inventive in- 
genuity which forges the chain seems to be guided by pure 
caprice, we cannot honestly say that they show why the 
conclusion T is true; they only constrain us to admit that 
it is true. I have introduced this point because of its 
practical importance. Our ideal of knowledge and demon- 
stration no doubt is that we should deduce each given 
proposition T from the determining grounds by which it is 
in fact determined in such a way as to explain //, and not 
simply assure ourselves of its certainty by a logical device; 
and if this problem is to be solved, it can only be by a 
direct progressive proof of this form. But it is soluble only 
within narrow limits, and where it is not soluble, where 
therefore we must content ourselves with the mere certainty 
of T) this form of proof by subsumption has not the least 
advantage over other forms. It is mere pedantry on the" 

312 THE FORMS OF PROOF. [Book 11. 

part of the logician to wish in spite of this to enforce it and 
when a proposition can be conclusively proved in two words 
by an indirect method to look about for a direct deduction, 
which can only be effected by a chain of arbitrarily selected 
links, which makes it a longer business to get to that 
certainty, and which does not in the least help us to see the 
reason why it is so. 

207. A second directly progressive method of proof is to 
start from the given proposition T, assuming it to be valid, 
and proceed to develop its necessary consequences. If 
among these consequences we find even one which contra- 
dicts either established facts or recognised general truths, T 
does not hold good as a universal proposition, and the proof 
becomes a mode of refuting a given proposition ; it then 
includes, as may easily be seen, that preliminary procedure 
above mentioned, by which we assure ourselves before 
entering upon the actual proof that among the given cases 
there is no contradictory instance against the validity of the 
proposition to be proved. If the development of the 
consequences of T however far it be carried discloses 
nothing inconsistent with known facts or truths, we have 
not even yet got enough to establish the truth of 7J for the 
next step in that development beyond the point at which 
we have stopped, might reveal the existence of a contra- 
diction hitherto concealed, but at any rate this procedure 
suffices in the field of science to recommend a hypothesis, 
which is then reserved for further examination. But the 
true province of this method lies in practical life : it is the 
method we employ to recommend proposals, arrangements 
that are to be adjusted, resolutions that are to be adopted. 
And here the incompleteness of the development of the 
consequences is no obstacle ; in all human affairs it is 
enough to ascertain what effects will follow from the appli- 
cation of a proposed measure within such a limited time 
and in such a limited field as we can readily survey : he 
-who wishes to take count of all the subsidiary effects which 


a microscopical examination might disclose, all the conse- 
quences centuries hence of what we do to-day, is a super- 
cilious pedant ; fresh measures will be taken to avoid 
minor disadvantages, and the remote future must take care 
of itself. 

208. A third form, the first directly regressive form of 
proof, proceeds from the assumed validity of 7"and works 
back to the conditions under which this validity is possible. 
The difference between this form and that just discussed is 
not considerable, but there is a difference : it is not con- 
siderable because the conditions requisite for the validity of 
T can only be found by taking T as their basis and de- 
ducing them as consequences from it, a procedure which 
coincides with our previous direct progressive method : but 
we see that there is a difference when we consider the 
nature of that which is thus deduced. We may take as an 
instance of both forms at once the ordinary way of solving 
a problem in mathematics ; for every such solution is at the 
same time a proof of the solubility of the problem, i.e. of 
the validity of the combination of ideas contained by the 
proposed problem T. If we assume that T is true and 
develop the consequences which flow from it, these conse- 
quences may be of various kinds ; some of them will be 
particular circumstances which agree or disagree with given 
facts, others will be general relations between various 
objects which are either consistent or inconsistent with 
truths otherwise established. If we only come upon 
particular consequences which disagree with given facts 
or secondary conditions, we may with certainty infer that 
T does not hold, though we do not see the reason why it 
docs not ; if T is a practical proposal, it may be that it is 
quite acceptable in itself and that it is only its execution 
that encounters some obstacle; and then we should have a 
case of the second form of proof: if on the other hand we 
come upon absurd general propositions which must be true 
if 7'is to be true, then besides the certainty that T'is im- 


possible we get also a strong hint as to the reason why it is 
impossible ; that reason must lie in the general truths which 
conflict with the absurd conditions we deduced, and herein 
we find what this third form of proof does for us. It npt 
only clears the ground for the subsequent discovery of a 
direct and progressive proof of the contrary proposition, 
but gives us a remarkably conclusive and palpable negation 
of a given proposition T in the disclosure of all the absurd 
assumptions that would be necessary if it were true ; and 
on this account this regressive proof is often preferable to a 
progressive one. 

It cannot establish anything but the falsity of T, and so 
remains a form of refutation. If in working backwards 
from T we come upon none but admissible conditions, we 
cannot infer that T is true except in mathematics ; for only 
in mathematics is it possible to develop from a proposed 
problem all the conditions necessary to its solution ; in 
other cases we can never be certain that we have really 
deduced from T everything without exception that is im- 
plied as a condition necessary to its truth ; the next step we 
took might bring to light an absurdity that we should have 
to assume. Affirmatively then this method is in matters of 
theory only able to establish the probability of T\ in prac- 
tice however we use it to recommend a proposal just as 
much as the foregoing progressive method. For when we 
want to secure the acceptance of a proposal we not only 
point out the consequences to be expected, but also show 
that the conditions of its execution are not incompatible 
either with the general requirements of justice and morality, 
or with the means which are actually at our command. A 
political measure always needs to be justified in these two 
ways, after the former method by its useful consequences, 
after this method by the admissibility, in the view of justice 
and morality, of all that it implies : and in our daily life we 
must take count not only of the advantage to be expected 
from a provision, but also of the price wfe must pay for it. 


209. A fourth method, the second direct regressive 
method, starts from given propositions and proceeds to 
prove from them the validity of T as the condition of 
which they are the result. This is a line of thought which 
we are very constantly impelled to follow : for the greater 
part of our knowledge of general laws is won in this way by 
reasoning back from given facts to that which must be 
assumed as the condition of their possibility. It is easy to 
see however that its most important applications belong to 
the method of discovery which tries to elicit from that which 
is given a T 7 which is as yet unknown. When the general 
proposition T is given and we are looking about for the 
several propositions which may serve to confirm it, the 
proper method is always to begin with the progressive de- 
velopment of that which as consequence of 7"must be true 
if T be true : only when we have made a comprehensive 
survey of these consequences do we proceed to compare 
the result obtained with experience or with other truths, in 
order to reason regressively from the truth of this result to 
the truth of 7! 

I will therefore postpone the consideration of much that 
might be introduced here, and will only mention one species 
of this method, viz. that which infers the universal truth of 
^from its truth in particular instances, complete induction 
or the collective proof. We are often compelled to employ 
it : it is not always possible to prove at one stroke that a 
proposition T holds good for all quantities, integral and 
fractional, positive and negative, rational and irrational, real 
and imaginary magnitudes ; but each of these several kinds 
of quantities may offer some special point of attachment for 
a proof that T is true of it ; if then we are sure that we have 
included all possible cases of T, that is in this case if we 
are sure that there is no conceivable kind of quantity besides 
those named, then we know that T is true of all quantities 
whatsoever. The general conception of quantity will then 
no doubt contain some reason for this universal validity; 


nevertheless we cannot always point out this reason, or at 
least we cannot always make it quite clear and self-evident ; 
and then we must have recourse to the collective proof. 

210. The necessity of including without any omission all 
the kinds of cases to which T can apply if T is to be proved 
universally true leads here to an interesting special form of 
this proof. Mere completeness of course can always be 
secured by dividing all the cases into say Q and non-<2, the 
non-<2 again into R and non-7?, and so on as far as we like, 
stopping say at U and non-7: but this is seldom of any 
use ; for even if we easily find separate proofs for the posi- 
tive kinds of cases QR 7, it is very difficult to find one for 
the negative remainder non-6 7 " which embraces a miscella- 
neous crowd of different cases. We are constrained there- 
fore to take a case Q, for which we happen to be already 
able to prove that T is true, and try to derive the other 
cases R U . ., etc. from Q in such a way that it may be 
evident that the changes by which Q passes into J?, and R 
into U, either do not affect the conditions which made T 
true in the case of Q y or else constantly reproduce them. 
This is the method, familiar to mathematicians, first formu- 
lated by Jacob Bernoulli^ of proceeding from n to n -f i, 
chiefly applicable when the several cases in all of which T 
is to be true form of themselves a series in which each suc- 
cessive (n 4- i) th member is formed in the same precisely 
definable way out of the preceding n^ member. If then it 
follows from the way in which the member n + i is formed 
from the member n that jTwhen true of the latter must be 
true of the former also, it follows for the same reason that 
it must be true of the member n -h 2, and so on for every 
member of the series. For instance in teaching the elements 
of algebra this method is usually employed to prove the 
binomial theorem for integral exponents in a palpable way 
by repeatedly multiplying the binomial into itself. 

The general idea of this proof however is by no means 
confined to mathematics, but is very often applied in com- 


mon life, sometimes under the not quite appropriate name 
of a proof by analogy. In support of a plan or a statement 
we first mention an instance in which the plan is evidently 
advantageous, the statement obviously true \ then we show 
that the other conceivable cases are in reality distinguished 
from this case by no feature that could possibly make a 
change in this respect ; and thence we conclude that T 
holds good universally. It is easy to see how a careless or 
sophistical use of this method may lead to error. Between 
two very different cases A and Z we insert a great number 
of intermediate cases, each separated from the next by an 
inconsiderable difference d. Then instead of showing that 
if Z'is true of A it must also be true of A+d> which is B, 
we assume that it is so because d is so trifling ; we reason 
similarly from B to C, and finally transfer the validity of T 
from A for which it really held good to a Z which by the 
accumulation of the many disregarded differences d has 
become entirely unlike A and does not in the least belong 
to the field to which T 7 actually applies. 

211. The indirect methods of proof may be treated more 
briefly. They bear formally the same relation to non- 7* 
that the direct methods bear to T, and the only circum- 
stance that makes them in some degree peculiar is that we 
wish to arrive by them not at non- 7 1 but at T: they are 
therefore not affirmative but negative proofs in respect of 
non- T. The fifth method of proof, the first indirect pro- 
gressive method, would have to show that non- T is false on 
general grounds, and this may be done by syllogisms in the 
first and second figures with a universal negative premise. 
But we shall seldom find an opportunity of applying this 
form of proof : if there be a direct proof for T we shall 
prefer it ; if there be none, a universal refutation of non- T 
is usually no easier. 

The only form of this method therefore which is prac- 
tically important is the secondary form, which in the place 
of non-7 7 , the contradictory of T, substitutes the complete 


sum of all its contraries. As these contraries are all quite 
definite positive statements, there is more hope of being 
able to disprove each upon general grounds, and therefore 
by a progressive method. The proof that non-7"is univers- 
ally false which is formed by the union of these several 
negative proofs is then evidently a regressive argument 
corresponding to the positive collective proof. When T 
and all the contraries of 7 T are conceived as together form- 
ing the sum of all possible relations which can subsist 
between S and P, the subject and predicate of T, the form 
of proof of which we are speaking becomes that which is 
known under the name si proof by exclusion: the truth of 
.7' then follows from the falsity of all the other members of 
this complete disjunction. One of the most important 
applications of this form is the special case of a tripartite 
disjunction, in which T has two contraries, i.e. in which 
non-7 7 divides into two contradictories : then we get the 
proof by the method of limits. We are familiar with this 
proof and its very great importance in mathematics, where 
it belongs equally to inventive and demonstrative reason- 
ing : every magnitude a is either equal to or greater or less 
than another magnitude d with which it may be compared : 
if it can be shown that it is neither greater nor less than */, 
the proposition a = d is proved. In practice this train of 
reason generally takes another line: for the above statement 
presupposes that our attention has already been directed to 
the definite magnitude d which is proved in the end to be 
equal to a. As a rule this is not the case, but we only 
know that a is less than a second magnitude b and greater 
than a third c : if then we can succeed in showing that the 
same relation constantly holds as we diminish the value of 
b to j3 and raise the value of c to y, the value of a must lie 
between two limits fl and y which are constantly approaching 
each other, and it will be possible to calculate this value 
with an approximation to the truth which may be carried as 
far as we please. The best known and most elementary 


example is the determination of the length of the circum- 
ference of a circle by enclosing it between a larger cir- 
cumscribed and a smaller inscribed polygon, and diminishing 
the former and increasing the latter without limit by continu- 
ally adding to the number of their sides. Such forms of 
proof deserve our attention ; they are the potent instru- 
ments by which we actually enlarge our knowledge ; the 
development and application of this method by Archimedes 
is a greater advance in applied logic than any that ever pro- 
ceeded from the merely syllogistic art of Aristotle. 

212. A sixth method, the second indirect progressive 
method, would begin by assuming non-7} and proceed 
to develop its necessary consequences, and then from 
their falsity infer the falsity of non-7] the last step of 
course being regressive. I will here refer the reader back 
to the second direct progressive proof, and only add with 
reference to this indirect method that it does not matter 
how many true propositions may be deduced from non- T: 
for it is quite possible for a number of true inferences 
to flow even from a false proposition with respect to points 
whose mutual relations are not affected by the error : but 
a single false proposition which follows as a necessary 
consequence from non- T does away with its universal 
validity. If this consequence merely conflicts with given 
facts tl^ere is properly no reason for calling this proof a 
deductio ad absurdum, though the name is sometimes given 
to all applications of this method : all that has been done 
is to prove that an idea which in itself is not unthinkable 
nor absurd is as a matter of fact untrue. But again absurd 
or nonsensical is strictly speaking not that which is known 
to be impossible in thought, but that which conflicts with 
all probable suppositions, with our general feeling as to 
what is true, and a number of truths involved in that 
feeling, provable perhaps but not yet actually proved. 
That 2 = 3 is more than absurd, it is impossible; but that 
the whole world is a thoughtless jest, that parents should 


obey their children, that we should reward criminals and 
be tender to sin, are absurd assertions. I would therefore 
apply the name deductio ad absurdum only to the indirect 
progressive proof which develops from non-7 7 consequences 
which are not impossible in thought, but which are in- 
consistent with a host of convictions accepted as truths 
and sufficiently established. This kind of proof is very 
constantly employed in daily life, especially whenever 
non-7 7 states a thought, which is perhaps in itself correct, 
in too general language, i.e. when it proceeds from too 
wide a definition of the subject S to which a predicate 
P is to be attached, or from too wide a definition of 
this P. It is in this way that we prove the unreason- 
ableness and foolishness of a proposed law, whether it 
gives or takes away rights and duties, by showing what 
further intolerable and monstrous consequences would 
follow if the proposal were carried out universally. Usually 
however the deductio ad absurdum is made to include 
also that form of indirect proof which deduces impossible 
consequences from an assumed proposition and thereby 
refutes it. 

A particular case of this is when the development leads 
to a consequence which at once does away with the pro- 
position from which we started, so that the inner con- 
tradiction which lay in the assumption of its truth of itself 
forces us to infer that it is false. As a simple instance 
we may take the indirect proof of the proposition T that 
on a straight line a b in the same plane and at the same 
point c only one perpendicular c d can be made to fall. 
Non-7 7 then would assert that several perpendiculars were 
possible at the point c under the same conditions. Now 
assuming that this is correct, assuming further that c d is 
the first perpendicular, i.e. that it makes with a b two 
adjacent equal angles a, any second perpendicular c e must, 
in order to be distinguished from c d, make with it at the 
point c some angle 6, while at the same time in order to 


be perpendicular to a b it must make with it equal adjacent 
angles. A look at the figure then is enough to show that 
the two angles #-fdand a 6 must be equal, and each 
equal to a right angle : but if a 4- 6 be a right angle, cr, 
being a part of this right angle, is not itself a right angle, 
which contradicts the original supposition that a is a right 
angle. The equation -f- d= a~ d can only hold good when 
6 = 0, i. e. when c e and c d coincide. The proposition T 
therefore holds good : at the same point in a straight line 
there can be only one perpendicular in the same plane. 

We are constantly led to proofs of this kind when we 
have to do with the simplest fundamental perceptions or 
propositions concerning a coherent field of thought : the 
impossibility of apprehending the relation of S to P other- 
wise than as it is expressed in T, i.e. the fruitlessness 
of the attempt to affirm non-TJ will always betray itself 
by the fact that the consequences which follow from it 
destroy or alter the subject S or the predicate P, which 
were both assumed to be valid for non-7 7 in the same 
sense in which they were valid for T. 

213. The indirect proof, like the- direct, admits of two 
regressive forms : these two, the seventh and the eighth in 
our survey, have but little to distinguish them ; they bear 
just the same relation to the falsity of non-7 7 that the 
two direct regressive proofs bear to the truth of 7 7 . 

The former (the seventh) method would work back from 
non-7 7 to the conditions necessary to its truth, and then 
reason back again from the falsity or inconceivability of 
these principles to that of non-7 7 . In its application this 
method differs but little from the corresponding progressive 
method; for the principles which are necessary to the 
truth of non-7 7 can only be found by taking non-7 7 as their 
basis and developing them from it as its consequences, i.e. 
progressively. The latter (the eighth) method would start 
from given facts or principles and proceed to show that 
they cannot be founded upon non-7 7 as their basis, but 


322 THE FORMS OF PROOF. [Book n. 

rather expressly require the falsity of non-7! This also 
we shall find can only be carried out by either developing 
non-7 7 progressively into its consequences, and ascertaining 
that if they held good they would make the given facts 
impossible, or by taking these given facts for basis and 
deducing from them, progressively as before, their necessary 
presuppositions : but this will very seldom be of much 
use, for in that case it will usually be easier to establish 
directly that T as such a presupposition must be true, 
than indirectly to establish that non-7 7 cannot be true, 

I will conclude this survey with the general remark that 
I believe that I have correctly distinguished in my classi- 
fication the various aims of demonstrative reasoning, but 
that not every one of these aims has corresponding to it 
an equally important and equally peculiar form of proof, 
clearly distinguishable from all the other forms; it was 
enough therefore to examine in detail only those which 
have in practice shown themselves to be methods that 
are frequently applicable. 

214. The reader will be surprised at the absence from 
my list of the proof by analogy, I do not believe in its 
existence. In all cases where we believe we can prove 
by analogy, the analogy in fact is distinctly not the ground 
of the conclusiveness of the proof; it is only the inventive 
play of thought by which we arrive at the discovery of a 
sufficient ground of proof : it is upon this ground, by means 
always of a subsumption of the individual under a uni- 
versal, that we establish the necessity of the proposition 
to be proved. Although it will take a considerable space, 
I think I must consider this point in detail. 

It may be regarded as a fundamental principle of analogy 
in the strict sense, holding good in all cases without 
exception, that of like 1 things under like 1 conditions like 1 
assertions are true, a statement which the mathematician 
further expresses in a number of special ways adapted 
1 [The German word is ' gleich' not ' ahnlich/' See note p. 327 below.] 

Chap, IV.] ANALOGY. 323 

to his various problems. It is easy to reduce this principle 
to the principle of subsumption : if P is true of S under 
a condition X, S and X may be comprehended in a general 
conception M, of which as such P is true ; under the same 
M we may subsume any other S which is like the first S 
and subject to a like condition X; therefore the same 
predicate belongs to this S as to the first. This trans- 
formation, which may here seem arbitrary and superfluous, 
cannot be dispensed with in the case of the second prin- 
ciple, of unlike things under like conditions unlike as- 
sertions are true. We may be inclined to regard this 
also as unconditionally true, but difficulties thicken upon 
us when we try to apply it. Suppose that unequal mag- 
nitudes a and b are divided by the same third magnitude 
c ; in this case the principle will hold good ; the quotients 
will be unequal. But take a second case : divide each 
of the unequal magnitudes by itself, and the principle seems 
to fail; the quotient in both cases is i. Of course it 
will at once be urged that the condition X y to which the 
unequal elements a and b are subjected, is just not alike 
for both; for when we divide each magnitude by itself , 
we introduce the inequality again into the meaning of the 
condition which was to have been alike for both. But 
this explanation will not cover the following third case; 
multiply both by o, and the product in both cases alike 
is o. It cannot be denied that the operation of taking 
a magnitude no times has but one meaning, and does 
not as m the former case depend upon the value of the 
magnitude to which it is applied : on the other hand it 
may be remarked with justice that in this case the meaning 
of the like condition or like operation X is precisely of 
such a peculiar kind as to annul the inequality of the 
magnitudes to which it is applied. Take a fourth case; 
if we square the unequal magnitudes a and b> the mean- 
ing of the condition to which we subject them is again 
dependent upon the magnitudes themselves as in the 

Y 2 


second case, only with the opposite result; the squares 
a 2 and 2 are unequal. Fifthly and lastly the results are 
once more equal, for both =i, if we raise a and b to 
the o th power. In this case the condition to which we 
have subjected the unequal magnitudes a and b seems 
to be independent of their value ; but in fact the raising 
to the o th power is a quite inconceivable operation; we 
must remember that in general a m ~~ n is merely another 

a m 
expression for ? and that accordingly a l ~~^ ? which is equal 

to a Q > is identical with-? and therefore this fifth case is 

identical with the second. If we wish to avoid all these 
ambiguities the only way is to say that of unlike things 
under like conditions unlike assertions are true when the 
condition is of such a nature as not to affect the unlikeness 
of the unlike things : but that like assertions are true 
of them when the condition is such as to annul their 
unlikeness. But these two propositions are mere barren 
tautologies : they do not enable us to decide even so 
much as whether the assertions to be made will be like 
or unlike without a previous analysis of each case to 
teach us what is the general rule M P under which 
a and b are really to be subsumed here, and what are 
the definite predicates P l , and P 2 which attach to them 
in virtue of the special sense in which they, as unlike 
kinds of M, partake of this universal P. When we have 
found these predicates P l and P 2 we see whether they 
are like or unlike; it is not by analogy therefore, but 
entirely by subsumption that the conclusion is arrived at. 

215. To the third principle, that of like things under 
unlike conditions unlike assertions are true, a higher value 
may be assigned ; it would in fact be inconsistent with the 
law of identity if an identical subject under really different 
conditions showed no trace of the influence of this differ- 
ence, and I shall have occasion some way further on to 


make use of this proposition as a not unfruitful maxim 
in the treatment of philosophical problems. But for the 
present what strikes us is the number of apparent ex- 
ceptions. How could the engineer solve the problem of 
constructing a machine which under changing conditions 
regulates itself and maintains a uniform motion, if the same 
subject or material substratum under different conditions 
absolutely must exhibit different effects ? A closer exami- 
nation removes this objection ; it teaches us that in the 
cases here concerned either the unlike conditions are not 
simple but go in pairs, or that the like subject is not simple, 
but a whole of various parts. But two pairs of conditions 
may with regard to a definite effect be equivalent, because 
the unlikenesses of the several members, in virtue of the 
definite relation which subsists between them, annul one 
another till the remainders are like ; on the other hand 
various unlike conditions may so work upon the various 
parts of a whole that the several effects in each case modify 
one another till the resulting state of the whole is like. A 
simple body which is out of all relation to others can never 
receive under the impulse of a force a the same motion 
that it receives under the impulse of a force b unequal to a. 
But under the simultaneous influence of a and b it may be 
moved at the same speed and in the same direction as under 
the combined influence of c and d: if these four forces operate 
in the same straight line, the equality of their algebraical 
sum, i.e. the condition that ad = cd, is enough to give 
a like motion to the body; or in more general language, 
every motion m may be conceived as the resultant of a 
countless number of different pairs of components. 

Now this result may be exhibited in various ways. If we 
regard the sums a b and c d as the conditions to which 
the body is subjected, then the conditions themselves are 
like one another, and the case comes under the principle 
that of like things under like circumstances like assertions 
are true : but if we leave the several forces separate, the 


case seems to make an exception to the third principle. 
Nevertheless I should like to maintain that this third 
principle is universally true ; for its true meaning plainly is 
that the sum of all the effects experienced by the same 
subject or substratum under different conditions will always 
be different. And so even if two pairs of conditions are 
equivalent in respect of one kind of effect which they 
produce on the same subject, it does not follow that they 
are also equivalent in respect of all their effects, and it is 
not proper to attend to the former only and neglect that 
part of their effect which is unlike. If a and b work upon 
a body in opposite directions, and c and d also in opposite 
directions, and if their sums or differences a + b and c d 
are like, the body certainly experiences the like motion m, 
and remains at rest if a b and c-=d- } but it obviously 
experiences very different pressures according as it is two 
large or two small forces that hold it in equilibrium. 
Though a self-compensating machine continues to act alike 
under constant and under varying conditions, yet the posi- 
tion of its parts changes as the conditions change, and it 
wears out faster when it is obliged to exert its compensating 
powers than when it leaves them unused, the conditions 
remaining uniform. If full sunlight falls upon one scale 
of a balance suspended in a vacuum, while the other is in 
shadow, the equilibrium is not disturbed, but the first scale 
is warmed and expanded more than the other. Lastly if we 
multiply a first by a b and then by b a, these conditions are 
certainly quite equivalent in respect of the magnitude of the 
resulting product, but not in respect of its structure, and 
a a b is in any case a different combination from aba. It 
would be easy to add to these examples, already sufficiently 
various, and thus to confirm the universal truth of the third 
principle ; but after all it is but of very little use for a proof 
by analogy; it never enables us to establish what all analogy 
aims at, viz. that in a second case the same thing happens 
as in a first, but only brings us to the negative conclusion, 


that any difference of the conditions in the same subject 
makes the likeness of the total effect impossible ; what is 
still like in this effect, and what unlike, we can never tell 
without an enquiry of another kind. 

The fourth principle needs but the barest mention ; that 
of unlike things under unlike conditions unlike assertions 
are true is, after all that has just been said, so evidently 
unfounded or ambiguous, that no useful application of such 
a statement is conceivable. 

I will only add in conclusion that the trains of thought to 
which the title of proofs by analogy is supposed to be 
appropriate do not even proceed directly from these prin- 
ciples, though they must be traced back to them. The 
presupposition on which they rest is rather that of similar 
things under similar circumstances similar assertions are 
true. Now similarity 1 is always a mixture of identity 2 in one 
respect and difference in another ; if therefore it is difficult 
to base any valid inference upon the foregoing propositions 
which separate the mingled elements, it is still less possible 
to do so when the two are indiscriminately fused together 
in the resemblances to which appeal is made. I think 
therefore that I have sufficiently shown that there is no 
such thing as a proof by analogy ; though I do not by this 
intend to deny that the observation of even remote resem- 
blances is of great assistance to the discoverer both in 
detecting new truths and in finding grounds for proving 
given truths ; for, to sum up my meaning briefly, there is 
no need to impugn the abstract validity of these three 
principles, but only their fruitfulness for demonstration. 
We cannot on the ground of the unanalysed similarity of 
two subjects transfer the predicate of one to the other, but 

1 ['Aehnlichkeit.'] 

2 [' Gleichheit/ It is impossible to adhere to a single rendering for 
* gleich.' Thus * unlike ' applied to magnitudes as on p. 323 might mean 
heterogeneous ; ' ungleich ' is therefore rendered there by * unequal ; ' 
but in the rest of this passage by * unlike.' Cp. Metaphysic, sect. 19, 


dhly on the ground of their demonstrated identity, identity 
at least in respect of the conditions upon which the predi- 
cate in question everywhere depends; and this always 
brings us back to setting down a universal proposition 
M P and subsuming both subjects under the determining 
conception M. 

216. We have still to consider those mathematical argu- 
ments which are commonly called proofs by strict analogy. 
As the name analogy originally meant proportion, every 
procedure that leads back to proportion has a reasonable 
claim to the title ; the effect of common usage however is 
such that when we hear of an inference by analogy we 
expect an argument which reasons directly from similars to 
similars, without needing to take a circuitous route through 
a higher universal. But the methods employed by mathe- 
maticians cannot be thus opposed to proof by subsumption. 
A proportion between four determinate magnitudes, a\bc\d, 
is merely the expression of a fact ; it only becomes a source 
of fresh inferences when the last two members are left 
indeterminate ; but in this form, a : b = m : n, it is the ex- 
pression of a universal law ; it asserts that the magnitudes 
yielded by the problem now before us at the moment are 
connected together in pairs in such a way that in every pair 
one member is to the other as a : b. If we give any definite 
value to m and n we get a syllogism in Darii, all the pairs 
of magnitudes which the problem yields (M) have the ratio 
P y viz. the ratio a : b \ but m and n (the S or subject of the 
minor premiss) are such a pair ; therefore m and n are to 
one another in the ratio a : b. No doubt this reduction to 
the first figure is very tedious ; but we deceive ourselves if 
we fancy, because of the shortness of the formulated ex- 
pression which the nature of the subject-matter makes 
possible in mathematics, that the train of thought also in 
a simple proportion is something shorter than that here 
stated. Even the simplest example of the rule of three 
is worked in this way. We say, if i' pound costs two 


thalers, 10 pounds cost 10x2 thalers : here we assume, 
what seems to us self-evident, that the ratio between the 
quantity of the article and the price is always the same ; 
accordingly we take the ratio of the one pound to its price 
as a general rule and bring the ratio of the 10 pounds to its 
price under it as a particular case of the rule : but the 
dealer perhaps sells the 10 pounds for 18 thalers and 
thereby shows that what we assumed is not self-evidently 
true in all cases, but that we really had to make the 
assumption for the purposes of our calculation : further it 
is evident that we tacitly conceive m and n as standing for 
quantities of the same article and of the same unit of 
currency as a and ^, and so in this respect also take the 
first case as the general rule and subsume the second case 
under it. Every general equation which exhibits one and 
the same content under two different forms is equally a 
general rule, which holds good only for that kind of magni- 
tudes which, by a convention which finds no expression in 
the formula itself, we intend to denote by these particular 
letters, and for which we originally showed the equation to 
be valid. It is not allowable therefore to substitute for 
the magnitudes m and n which occur in an equation any 
other chance magnitudes M and v, and to regard the equation 
as still valid : we must know beforehand that //, and v can 
be subsumed under the species m and n of which the 
equation has been proved to be true. Suppose we have 
proved by actual multiplication and by the argument from 
n to n + i that 

that does not give us the right to infer also that 

(--- 1) 

\m / o 


i.m mi. 2 
for in the first formula m stood only for the class of positru 

330 THE FORMS OF PROOF. [Bookil. 

wLole numbers, for which alone the proof by multiplication 
was feasible, and a fraction cannot be subsumed under it. 
If on the other hand we had found means to prove in the 
first instance that the binomial theorem in the first case holds 

true for the fractional exponents ? whatever positive value 

may be assigned to m and #, we might have deduced the 
first formula directly from this, since every whole number 
m may be expressed in the form of an improper fraction. 

217. In conclusion I should like once more to connect 
what I have said with the dictum de omni et nullo or the law 
of disjunction. If S l and S 2 are two species of the genus 
M or two particular cases of the universal M^ and if P may 
be predicated universally of M^ we know that P may be 
predicated of S l and *S 2 not in this universal form but in 
the modified forms P 1 and P 2 . Now in a special case it 
may happen from the way in which the various predicates 
P Q R are connected in M, that the various groups of 
characteristics / 1 q x r l , / 2 q* r 2 , ^ q z r z which they form in the 
several subjects s l s 2 s^ must be identical with one another ; 
they then make so to say a secondary predicate O, which 
may be ascribed to M itself, and which equally attaches 
without modification to every species of M. Thus the con- 
ception of the triangle M requires three angles pqr, but 
the various values of these angles in the various kinds of 
triangles always make up the same sum II = 2 right angles ; 
this identical characteristic IT therefore attaches to all 
triangles and we may at once ascribe it to any single 
triangle when we have simply subsumed it under its genus. 
But apart from such special cases the / 2 or q* that will be 
proper to an s* remains indefinite, with the single limitation 
that it must be a kind of Q, and that it must always be 
present, even though its value diminish to nought, in which 
case this nought must be capable of explanation. If this ^ 2 
is to be determined, there must be a rule according to which 
the specific peculiarity of S l (which makes it not only a kind 


of M but this particular kind) helps to determine the modi- 
fications of the general characteristics of M^ in this case 
the modification of Q ; and we must assume that the peculiar 
nature of S' 2 will follow the same rule in determining (f, the 
modification of the general characteristic Q which is appro- 
priate to it. If we know this rule we can determine ^ 2 , and 
this is precisely the case which is called inference by strict 
analogy, though as we have seen this rests upon nothing 
but the subsumption of a case under the like universal rule. 
But when this rule is not known, we still feel inclined to 
find out q 1 by considering the resemblances and differences 
in the relation of S l and S 2 to each other and to M, and the 
procedure based upon this we usually call inference by 
analogy; but it only enables us to guess the right result, 
never to prove it. It was known by the forty-seventh pro- 
position that for right-angled triangles the square on the 
hypotenuse h is equal to the sum of the squares on the 
sides a and b which enclose the right angle. As this 
relation can depend upon nothing but the general pro- 
perties of the triangle, the right angle, and the length of 
the sides, it is a quite justifiable impulse which bids us 
seek an analogous proposition about the square of the 
subtending side for other values of the subtended angle. 
If we simply put the formula in the general form // 2 =# 2 -}-/> 2 
there is t no longer any mention of the right angle \ but the 
formula we are seeking must mention the subtended angle, 
for it is evident at a glance that, a and b remaining the 
same, h gets longer as the angle increases and shorter as it 
diminishes. Accordingly to make the Pythagorean formula 
complete we must add another term which will become 
nought when the included angle </> = 90 : and as we cannot 
measure h by the angle itself, but only by a length de- 
pendent upon it, or by a numerical coefficient dependent 
upon it that determines another length, we may set down 
tentatively %'*= a?+ $* m cos (/>. The alternative sign + 
is seen at once to be needless when we reflect that when <J> 


increases beyond 90 h still increases but the cosine becomes 
negative; we only need the minus sign therefore in the 
formula. In order to find m which is as yet indeterminate 
we turn to the two limiting values of $, <f> = o and (/> it. 
In the latter case Ji l becomes equal to (a -f Vf and cos 
<j> = i ; in the former case ti* = (a frf and cos <$> = -j- i ; 
both cases alike give us h 2 = # 2 + ^ 2 2 # cos 0. Now 
this formula is in fact correct for all values of $, but it is as 
yet by no means proved ; it covers with certainty only the 

three special values of <, viz. $ ~ , </> = -TT, </> = -, from 

which it was obtained : it would be easy to find another 
formula, e.g. 

/& 2 =0 a + P- 2^COS<|>.COS 2 (<7T- (/>), 

which would also cover them; which of the two is also 
satisfied by all the intermediate values of <j> remains un- 
settled, till by an easy geometrical construction, with the 
help of the forty-seventh proposition, we decide that the 
formula we first took is universally true. I have dwelt upon 
this simple example in order to show how many subsidiary 
considerations are necessary before our efforts to discover 
new truths by the analogy of given truths can even be put 
into a path which promises success. 


The discovery of grounds of proof , 

218. IN any demonstration of a given proposition T 
the most important thing is to find the major premiss G, 
from which by appropriate subsumption T is to follow as 
necessary consequence. This problem, obviously a problem 
for the discoverer, does not admit of any logical rule by 
which the solution could always be found with certainty, 
without counting upon the free co-operation of the sagacity 
of the individual enquirer. We must suppose that previous 
reflexion has already supplied a number of general truths, 
which are related to the content of the given T in such a 
manner as to be serviceable for the purpose in hand, and 
which, recalled to consciousness by the similarity of the 
matter in question, suggest themselves to the seeker as 
grounds for explaining the given proposition. But over 
and above this he must have the keenness of mental 
vision which detects among these truths the appropriate 
ground of proof, and sees the changes which perhaps are 
necessary to the subsumption of the given proposition 
under it, and this we must allow is to a large extent matter 
of native talent and not even independent of the moods of 
the moment. The logical relation however which subsists 
between the parts of a true and therefore demonstrable 
proposition must be able to give us at any rate such a clue 
as may save us from groping entirely in the dark and to 
some extent put us into the way of finding, after further 


starch of course, the ground of proof. This clue lies in 
nothing else than the . fact which we remarked some time 
ago that every true universal proposition T, when we sup- 
plement and complete its subject and its predicate by all 
the subsidiary characteristics which are hinted at or implied 
though not expressed, must become an identical proposition. 
If then for the conception S, which occurs as subject in the 
proposition T^ we substitute this completed sum of the 
several ideas which it contains in the forms of combination 
proper to them, this must include the ground which justifies 
the predicate ; on the other hand if we substitute for P in 
its completeness the sum of the several ideas included in it, 
this must include all the requirements which the subject 
must satisfy in order that the proposition T may be true. 
I will attempt to illustrate by a few examples the use of this 
clue, and as discovery and proof here in fact follow the 
same road, I shall treat some of these examples as proofs 
of the given proposition T and others as instances of its 
discovery, i. e. of the solution of the question what relation 
expressible in a proposition T must subsist between S and P. 
219. Suppose first that we have to prove the given pro- 
position T, that the angle in a semicircle is a right angle. 
By analysis of the subject we find that by the angle in 
question we have to understand one whose enclosing lines 
start from the extremities a and b of a straight line a h and 
intersect each other at a point in the circumference of a 
circle described about a b as diameter. Now if the second 
part of this definition, which determines the position of the 
point of intersection <?, is to be satisfied, the distance of e 
from c the point which bisects the straight line a , 
must be equal to half this line, i. e. to a c or c b. 
This requirement which follows from the definition of 
the subject suggests at once the one slight subsidiary 
construction that we need : we must draw this line e t, in 
order to bring before our eyes the relations upon which 
depends the necessity of the given proposition T. When 


we have drawn e c the triangle a e b which we already had is 
divided into two isosceles triangles a e c and e c b, while the 
angle at e is divided into two angles a and /3 : from the fact 
that both triangles are isosceles this follows, and so far this 
alone, viz. that the angle e a c a and that the angle e b c = 
(3 ; but from the way in which these two triangles make up 
the triangle a e b, e c being common to both, and a c and c b 
falling in the same straight line, it follows further that the 
four angles a, a, /3, /3, are together equal to the sum of the 
angles of the triangle a e b. We have then 2 (a + fi) = two 
right angles, and as a -f ft is the required angle in a semi- 
circle, we have found that it is equal to a right angle. 

It is not always that we can get what we want by such 
an easy analysis as in this very simple case : let us therefore 
take another case to illustrate an artifice that is very fre- 
quently applicable. We may perhaps already have got a 
proposition T which teaches us what is true of a subject 
which is not equal to S the subject of the given proposition, 
but diverges from it by a difference that can be stated ; 
supposing then that by removing this difference we cause 
this subject to pass into the given subject S, and are able 
to show how the relation expressed by T is altered by this 
operation, we shall prove the given proposition T if it is 
true, or find the true proposition T if the given proposition 
is false, or if none were given at all.* 

Suppose the question to be what is the sum of the angles 
of a triangle. Assuming that the propositions concerning 
parallel lines and their intersection by a straight line have 
been established without taking triangles into consideration, 
we take two straight lines a d and b c parallel to one another, 
and intersected by a third straight line a b in the points 
a and b. These three lines thus form no triangle, but an 
unclosed space ; but we know S the sum of the two angles 
da b and a b <:, and know that it is equal to two right angles. 
If we now make the line a d turn about the point a so as to 
incline towards b c, there is formed between its new position 


and its old one an angle </>, which is taken away from S the 
sum of the interior angles ; but at the same time there is 
formed between b c and the line which has been deflected 
to meet it a new angle, the third angle which together with 
the remainder of S the sum of the original angles makes up 
the three angles of the triangle now formed, and which by 
the propositions about parallels is equal to the angle (/> 
which was excluded from S. Thus therefore in the passage 
to a triangle from what is not a triangle the sum of the 
angles contained by the three lines loses (f> and gains <p ; 
it is therefore equal to two right angles in the triangle as 

220. Suppose we want to prove or to find the conditions 
of equilibrium for a perfectly free and absolutely rigid body, 
operated upon at various points by various forces in various 
directions. In the conception of a body here employed 
perfect freedom needs no further analysis ; as absence of 
every conditioning relation to others it is quite clear as it 
stands ; only if the relations were present should we have 
further to determine their import : the absolute rigidity of 
a body means that the distance between any two points in 
it is unalterable. 

Now if no force were acting upon this body, we should 
be able to say of it that it either was at rest, or was con- 
tinuing an original motion at a constant speed c : we should 
therefore only have to set down c = o in order to express 
the conditions of the equilibrium intended, the equilibrium 
of rest. But in order to decide how the body maintains 
equilibrium when forces are acting upon it we must adopt 
the same method as in the preceding case and first see how 
it would move if it did move, and then negate all the con- 
ditions which would be inseparably bound up with this 
motion. This is not merely a useful contrivance without 
any logical basis ; for the equilibrium we are now seeking 
must be conceived not as mere rest but as the negation of 
fhe movements which tend to disturb it. * Now as the only 


kinds of motion are motion from place to place, rotatory 
motion, and thirdly the combinations of these two, all we 
have to do in order to determine the equilibrium of the 
body is to consider the conditions of the two first-named 
kinds of motion; negate them and the possibility of the 
third kind is gone. 

221. If we first consider only movement from place to 
place or movement of translation, expressly excluding all 
rotation, it follows from the definition of rigidity that all the 
parts of the rigid body must move onward in rectilinear and 
parallel paths and therefore with the same velocity. In 
whatever way a force acts therefore, if it has given to a, one 
part of the body, a velocity c, it must always, provided the 
movement be one of translation and not of rotation, have 
given the same velocity to , any other part of the body. 
Hence we are able, to our great convenience, in estimating 
the movement of translation which finally results from all 
the forces acting upon a rigid body to neglect the fact that 
they act upon different points : we may treat them all as 
acting, in lines parallel to their given directions, at an arbi- 
trary point in space, at which we suppose the mass of the 
body to be concentrated, and then by the known rules for 
the composition of forces determine the resulting movement 
jR which they would impart to this point ; the magnitude 
and dirg^tion of this resultant R are then identical with the 
magnitude and direction of the motion which the body re- 
ceives under the united influence of these forces, and it 
remains at rest when R = o. If we express this by saying 
that the body rests when the effects of all the impulses to 
motion which are brought to bear upon it annihilate one 
another, the proposition is an identical proposition for 
which no reason need be sought : our explanation however 
further states the condition under which that annihilation 
takes place, viz. the very same condition as that under 
which it will take place when all the forces are acting upon 
the same point. 




222. In mechanics however it is usual not to state this 

condition under this form It = o, but to break it up, for 
convenience in applying it to calculations, into three equa- 
tions, which I proceed to mention, since the feasibility of a 
logical precept is certainly one of the questions which applied 
logic ought to consider. If the number n of the forces 
acting upon the body be considerable, it becomes laborious 
to find the last resultant R by first of all getting a first re- 
sultant out of two of these forces, and then a second out of 
this and a third force, and so on till the last force is com- 
pounded with the last preceding resultant. Moreover the 
angles which the direction of each force makes with that of 
any other, and which would have to be considered in this 
calculation, are seldom included among the data originally 
given ; but where these data have to be first determined by 
the examination of a given state of things, it will be prefer- 
able here as elsewhere to characterise the directions of all 
the forces by their relations to a single common standard, 
instead of measuring the divergence between every two. 
The usual proceeding then is to lay down three axes X YZ, 
at right angles to one another, and then to determine the 
direction of each force P by the three angles a ft y which it 
makes with these axes or with lines parallel to them, at the 
same time conceiving each force as resolved into three 
components parallel to these axes, which forces will accord- 
ing to a familiar proposition be P cos , P cos /3, and P cos y. 
The three sums then made by adding together all the com- 
ponents of like direction, i. e. the sums 2 P cos a, 2 P cos /3, 
2) P cos y, will be the resulting forces which tend to move 
the body in directions parallel to the axes X Y and Z re- 
spectively : if each of these sums as they stand be equal 
to nothing, the body does not move from its place in 
any of these three directions, and therefore does not move 
at all, for any movement in an intermediate direction 
4 would include a simultaneous change of place in the direc- 
tion of two of these axes at least, and this has just been 


denied. So instead of R = o we have these three equations, 
2 P . cos a = o, 5 P . cos j3 = o, and 2 /> . cos y o, to 
express the condition which annihilates all movement of 

223. We have still to look for the other conditions which 
make the rotation of the body impossible. Suppose now 
that a straight line rotates about one of its points ; then 
with the exception of this one point which we regard as 
fixed (thus making it impossible for the whole line to have 
any movement of translation) all the other points of the line 
alter their co-ordinates. The line therefore cannot rotate if 
two of its points have unalterable co-ordinates. But though 
the line be fixed along its whole length, a plane which con- 
tains it may rotate about it : then all the points in the plane 
which lie outside this axis alter their co-ordinates : the rota- 
tion of the plane therefore becomes impossible if any point 
in it outside the axis be fixed, or in general if the three 
angular points of a triangle drawn anywhere in the plane be 
fixed. The same condition is obviously sufficient to make 
rotation impossible for a rigid body, every point of which is 
at an unalterable distance from every point in a fixed plane 
taken at will in it. The condition which prevents rotation 
therefore might be expressed by saying that the three an- 
gular points of a triangle drawn anywhere within the body 
do not alter their co-ordinates. But the proof that this con- 
dition was fulfilled would not be at all a convenient one : in 
order to prove it by applying the previous three equations 
to each of these three points we must be able to prove what 
is the resultant effect at each of them of all the forces acting 
not at this point but at other points : but this, as will easily 
be seen, is the very thing that we are still trying to ascer- 
tain. We must take another course therefore, and, since 
the position of the triangle just mentioned is perfectly arbi- 
trary, the course which most naturally suggests itself is to 
dispose its three angular points in the three axes X Y Z 9 by 
reference to which we have already determined the directions 

Z 2 


of all the forces in operation : but the position in each axis 
of the angular point which we place in it is also perfectly 
arbitrary: we may therefore regard every point in each axis 
as a point whose position is unalterable, i.e. we may regard 
the three axes themselves as three fixed lines, in relation to 
which, if rotation is to be excluded, no point of the body 
can change its position and distance. If finally we consider 
the three axes as three dimensions which lie within the body 
itself, or as identical in position with three series of points 
in the body at right angles to one another, it follows from 
the definition of rigidity that the fixity in space of these 
series of points is all that is required to make any change of 
place impossible to the remaining points of the body. The 
problem therefore reduces itself to showing that all the forces 
in operation are unable to impart a rotatory movement in 
any direction to any of these three series of points, or to 
any of the three axes X YZ now conceived as capable of 
moving out of their previous direction. 

224. This last way of treating the matter however would 
not serve as a convenient basis for calculation except when 
the directions of all the forces concerned passed through the 
three axes. This will not generally be the case : in order 
to take into account those forces which when produced go 
past those series of points without cutting them, we must 
substitute for the three lines three planes intersecting each 
other at right angles, each of which will therefore include 
two of these axes : the direction of each force produced if 
necessary must cut one of these planes. The problem now 
is to show that all the forces in conjunction are unable to 
cause either the planes X Fand X Z to rotate about X y or 
the planes Y Z and YX to rotate about F, or the plane 
Z Y and Z X to rotate about Z. Let us consider the con- 
ditions of rotation about Z. Any force P acting in any 
direction upon a point of the body whose co-ordinates are 
x y z, and making with the three axes the angles a ft y, can 
as before be decomposed into three forces P cos a, P cos /3, 


P cos y, parallel to the three axes. The last of the three?we 
need not consider here ; it could only cause a movement of 
translation in the direction of the axis Z, which is already 
excluded Jby the equations of 222, or a rotation of the 
plane X Y about Xor Y, which also need not be considered 
at present. Of the two other forces P cos a is perpendicular 
to the plane Z Fand Pcos (3 to the plane ZX; the two 
tend, as is shown by an easy construction, to cause the 
planes ZJfand Z F, and so the body in which these two 
planes are immoveably united, to rotate in opposite direc- 
tions : the direction of the rotation which actually results 
would therefore depend upon the difference between the 
two forces. Not simply upon their difference however, for 
a proposition which at present we will only allude to, teaches 
us that the rotatory effect of a force which is perpendicular 
to a line is to be measured by the product of its intensity 
into the distance of its point of application from the axis of 
rotation. For the force P cos a this distance is j/, and for 
the force P cos fi it is x : the difference of the products 
y P cos a and x P cos /3, or the difference between the two 
momenta, must be equal to nought if P is to cause no 
rotation about the axis Z. We must repeat the same con- 
siderations with regard to all the forces concerned, and we 
finally get, as the condition which prevents all rotation about 
the axis Z, the equation 

S ( y P cos a x P cos /3)= o. 

The other equations which make rotation about the axes X 
and F impossible, will obviously, as the three directions are 
perfectly homogeneous, be of the same form ; and, since 
even artificial aids to memory are not beyond the province 
of applied logic, I will remark that the equation for non- 
rotation about an axis never contains the elements which 
refer to this axis, but consists of the sum of the differences 
of two products, each of which unites a component force in 
the direction of the second axis with that co-ordinate of its 
point of application which is parallel to the third axis. The 


foimula 2 (z P cos /3 y P cos y) = o annihilates rotation 
about X ; the third formula 2 (# P cos y 2 P cos a) = o 
annihilates rotation about the axis K 

225. The proposition about the equilibrium of rotatory 
forces which we made use of in the preceding discussion is 
easily arrived at in the domain of statics by a slight device 
which reduces the question to the composition of motions. 
If I now select another mode of proof, I do so of course 
with no idea of improving the science of statics ; I only 
adopt a treatment which is as far as possible independent 
of all merely happy contrivances, in order to illustrate the 
way in which the grounds of proof are brought to light by 
the analysis of the problem itself. If the rigid line a , 
whose length we will call , rotates about its fixed extremity 
a, this implies that all its points describes a segment of a 
circle p <o with the same angle co and with a radius p, which 
for each point is equal to its distance from the point a. If 
now a force W acts upon the point , and causes , in 
whatever way, in the time t to pass through the segment 
n co, it must likewise have compelled any other point in the 
line at the distance p to describe in the same time / the 
segment p &> : and conversely, any force which applied at 
the point p has caused this point to move through the small 
segment p o>, has necessarily compelled all the other points 
in the line to describe segments corresponding to their 
distance from a. We now ask what must be the nature of 
the two forces P and Q in order that when they are brought 
to bear at the points / and q respectively they may produce 
precisely equal results, and accordingly when acting in 
opposite directions upon the line a b may prevent each 
other from making it rotate. Now the conception of 
rigidity, i.e. the conception of the simple immobility of a, 
is too far removed from conceptions of movements, to tell 
us how the latter would be affected by the former : we 
should have first to conceive rigidity itself as the result of 
movements, in order to make it homogeneous with the 


other movements upon which it is to exercise a restraining 
influence. Further it is impossible to compare P and Q so 
long as they act under different circumstances whose 
modifying power is yet unknown : we can only estimate 
them by velocities (f> and \|/ which they would impart under 
perfectly similar conditions to a perfectly similar moveable 
object : and lastly though P and Q may be applied at the 
single points p and ^, they cannot operate upon them 
alone ; in order to set up or to hinder a rotation, the effect 
of each must spend itself over all the points in the line a b, 
and we must know the mode of this distribution before we 
can understand how the effect of the one can annihilate the 
simultaneous effect of the other at every point in the line. 

226. These requirements we may satisfy in the following 
way. Suppose that a b, which is equal to #, is first of all 
a perfectly free rigid line, consisting of an infinite number 
n of homogeneous points which are compelled (how does 
not concern us) to maintain unchangeable distances from 
each other. Suppose that a number n of equal and 
parallel forces operate perpendicularly upon this line so as 
to give to each element of it the velocity oj ; then the total 
force W) equal to n o>, will urge the whole line forward, all 
the points moving in parallel directions. This movement 
of translation passes into a rotatory movement when we 
give to the various points of the line various counter- 
velocities, which must be conceived as at right angles to a b 
not only at the beginning of the rotation but at every 
subsequent moment. To the extremity a we assign a 
counter-velocity to by which it becomes the fixed point 
which our problem requires; to the point b we give a 
counter-velocity which = o, so that it maintains undimi- 
nished the velocity co imparted to it by W\ the intermediate 
points must meet so much resistance as will leave to each 
point p, whose distance from the fixed point is /), a residual 

velocity whose amount is already known, viz. the arc ~.o), 


whose length is to co, the path of the free end, as p is to n : 
the sum of the velocities of all the points />, from p = o to 

p = #, must be equal to Now a force P, which would 
give to a free element the velocity <p, would give to an 
element / in our rigid line the velocity - <, if / were sub- 
ject to the above-mentioned resistance but able to move by 
itself; but as it cannot move by itself, the impulse imparted 
to it must distribute itself over the whole line. However 
this distribution may be effected, we already know the 
result ; it can produce nothing but a rotation of the whole 
line, in which every point p receives a velocity proportionate 
to its distance from the fixed point and the sum of all the 

velocities is - Every point p therefore receives the 
2 n J r 

velocity - - - Precisely similar statements may be 

made about a second force Q, which would give to a free 
element the velocity \ff, but to an element q of the line 

which is fixed at one end would give the velocity ~-^\ 
when applied at q it would give to any other element p of 

the line the velocity - - - - Now if these two forces 
3 n \ji n\ 

operating at p and q or the two velocities produced by 
them are to be such that when acting in the same direction 
either would annihilate one and the same third movement 
of the line, or that when acting in opposite directions they 
would counterbalance each other, then for any point p the 
two expressions which we have just found for their effects 
must be equal to one another, and so therefore / <t> = q ty, 
and </> : ty = q :/. In other words the length of leverage 
must vary inversely as the strength of the force. 

227. The following would be a very plausible, and yet 
an inadmissible way of deducing the same proposition. 


Suppose that at the same point m of a lever playing in a 
vertical plane two equal forces P and Q are acting in 
opposite directions; it is self-evident that under these 
conditions equilibrium will be the result. Now if, as is 
commonly done, we imagine Q to be a weight, suspended 
by a hook or cord at m, and P as a strain exerted from 
above, we tacitly assume that it is indifferent~whether of the 
infinite number of infinitely thin perpendicular strips 
into which Q may be decomposed in thought each severally 
grapples the point of the lever which it would touch if 
produced, or whether all these several forces operate upon 
the lever only through a single representative which unites 
them all, viz. the cord. Once assume this, and it must 
also be indifferent whether we conceive Q as one body, or 
as divided perpendicularly by a geometrical plane into two 
halves which touch one another at the surface of section, 
and each of which is attached to the lever by a separate 
cord which unites all its forces in one resultant. If then m 
was the distance from the fulcrum of the original point of 
attachment, m x and m + x are the corresponding 
distances of the new points of attachment of these two 
cords. In other words equilibrium is preserved when two 

forces each of which is equal to - > and whose sum = P, 

are applied at equal distances right and left from the attach- 
ment ol the opposite force/*: for the cords themselves, or their 
tensions, are now the forces which are directly applied. Now 
so long as these tensions are the resultants of the forces of 

gravity united in the two bodies ? it is evident that it is 

quite indifferent how these bodies ~ are shaped in other 

respects, indifferent therefore whether they still touch one 
another as before, or whether by increase of their length 
and diminution of their thickness they become two separate 
bodies with a space between them. If we follow out this 


line of thought we see that it is quite possible to carry the 
displacement of one to the left and of the other to the 

r 2 

right by equal distances x as far as we please, till at last x 
becomes equal to m : when that is done one > say the one 

that was displaced to the left, has reached the fulcrum a, 
and no longer produces any effect upon the lever : the 

other has arrived at the distance 2 m from the fulcrum, 

and the equilibrium is still preserved under the condition 
.that P, which = Q, operates at the distance m from the 

fulcrum, while operates at the distance 2 m. 

But though this exposition brings the matter before us 
very plainly, it is nevertheless absolutely inconclusive. So 

long as x was less than m^ the that was moved away to 

the left had still a recognisable and intelligible influence 
upon the equilibrium of the lever; we could still see 
plainly that it together with the other half that was moving 
away in the opposite direction made up the force that was 
sufficient to counteract P\ but so soon as x becomes equal 

to m, and the effect of this altogether ceases, there is a 

break in the thought : for one of the points of relation has 
vanished, and our whole reasoning was founded upon its 
relation to the other. For when we first applied Q at the 
point m itself, and then disposed the two halves of Q 
symmetrically on either side of m, what we inferred held 
good in the first instance for the free line a b, which was 
supported at m by the force P : the fixing of the end a was 
not contemplated at all : though of course the same infer- 
ences held good also for the case when a was fixed, so 
long as it could be proved that, irrespective of this, equili- 
brium was maintained by the way in which the weights 


were distributed ; for if equilibrium was maintained thus vt 
could not be disturbed by the fact that a was over and 
above this regarded as fixed. But so soon as the influence 
of one half of Q vanishes, we no longer have equilibrium 
on the same grounds as before, and it is by no means self- 
evident that the vanished condition is exactly replaced by 
the fixing of the end a. We should in fact need for this 
special case to find a subsidiary proof which should show 
that a being fixed the effect of the half of Q was all along 
getting less and less as it approached a, and that equilibrium 
was nevertheless maintained; therefore it would continue 
to be preserved when the influence of this weight was 
reduced to nothing, while the other was removed to a 
corresponding distance. But if we examine it, we see that 
this subsidiary proof would in reality be the proof of the 
main question, i. e. it would be the proof of nothing less 
than the proposition that the power of equal forces to move 
a lever varies as their length of leverage. This mode of 
statement therefore, however plainly it brought the propo- 
sition in question before us, did not in the least prove it, 
but only assumed it in a circle which it is easier to recog- 
nise than to state briefly. 

228. Complicated mechanical problems cannot always be 
solved by directly compounding all the forces in operation 
so as to arrive at their final resultant ; we often have to state 
certain universal conditions which it must satisfy, or certain 
limits within which it must keep : with these assumptions 
then the several data of the given case supply means for the 
complete determination of the result. These methods, 
among which we need only mention the application of 
d'Alembert's principle, are quite invaluable and cannot be 
dispensed with : but as they do not clearly show the history 
of the result which we calculate by them, we still feel a wish 
to employ direct constructions so far as possible. I will 
mention in connexion with the preceding problem of the 
equilibrium of rotatory forces that of the motion which they 


generate when they are not counteracted. The rule for 
calculating it is reduced to these two very simple proposi- 
tions : (i) if a force acts upon a body that is able to move 
freely, its centre of gravity takes the same rectilineal motion 
which the whole mass of the body would take if it were 
concentrated at the centre of gravity and there acted upon 
by the force : (2) at the same time the body takes the same 
rotatory motion which it would receive from the same force 
if its centre of gravity were fixed. Now in this very neat 
division of the result there lies a paradox. For if the 
direction of the force passes through the centre of gravity, 
there arises according to the second proposition no rotation, 
but only a rectilineal movement of translation, and yet we 
should suppose that in this case the force was acting upon 
the body under the most favourable conditions : but if the 
direction does not pass through the centre of gravity, in 
which case the force would seem to act under less favour- 
able conditions, there follows not only the entire previous 
result but also a rotation, which strikes us as an addition 
without any obvious reason. If the compound velocities of 
the various parts of a body which is at once moving on- 
wards and rotating be decomposed into velocities in the 
direction of its rectilineal course and velocities in the 
directions perpendicular to this and to the axis of rotation, 
the sum of all the former components, each multiplied into 
its differential-mass, is equal to the product of the whole 
mass multiplied into its rectilineal velocity ; and we easily 
convince ourselves that when the body is at once rotating 
and advancing, though the several elements have various 
velocities in the direction of its course, yet the sum of all 
these velocities is neither increased nor diminished, but 
only otherwise distributed than it would be in the same 
mass advancing without rotating. But the other com- 
ponents remain, and though they have opposite signs for the 
two halves of the rotating body, yet they do not on that 
account annihilate each other: they are motions which 


actually occur, and we are forced to ask where they comr 

229. It is sufficient to answer this question in the 
simplest conceivable case. Let a and b be two equal 
masses, which we conceive to be concentrated at their 
centres of gravity : suppose that they act upon each other 
so as to. remain always at the same distance a b from one 
another : we may say then that a and b are united by a 
rigid unchangeable line a b which has no mass. In order 
to simplify the figure to be drawn, conceive a b to be so 
fitted into the angle of two rectilineal axes which intersect at 
O that a lies upon the axis X and b upon the axis Y: at 
starting then we have, for the mass <?, x = O a and_y = o> 
and for , x = o andj/ = O , while for the centre of gravity 
of the system a -f , which lies in the centre of the line a b t 

we have x = andj/ = We will now suppose that 

a certain velocity is imparted to the mass a in the direction of 
the axis X, and that a a is the path which it would traverse in 
an indivisible moment of time under this impulse if it were 
free. As no force is acting directly upon the mass , it would 
then remain at rest, and the line a b which expresses its 
distance from a which has moved away would be longer 
than the original line ab. But the forces in operation 
between a and b, which according to our assumption main- 
tain the distance a b unaltered, oppose themselves at every 
moment to the beginning of this elongation the measure of 
which would be a b - a b, and prevent it, by making the 
two bodies approach one another in the direction of the line 
at the extremities of which they would be found if the 
elongation actually took place. Since neither of the two 
masses can one-sidedly compel the other to follow it, but 
both masses, being assumed to be equal, must by the 
principle of the equality of action and reaction displace each 
other to the same extent, we shall find their new positions 
a 1 and fi by cutting off from the line a b the length a 1 equ4 


{ o , and from the line b a the length b /3 also equal 

to If from a 1 we let fall an ordinate, which we 


will call dy y upon the axis X, and from /3 let fall a perpen- 
dicular, which we will call dx^ upon the axis Y, we have two 
equal and similar triangles, and thus we get for a. 1 and /3, 
the two extremities of the now displaced line a l>, the 
ordinates dy and O b dy respectively ; and therefore for 
the centre of gravity, which is still the centre of this line, we 

have y = : but this was also the ordinate of the centre 

of gravity before any velocity was imparted to it : the centre 
of gravity therefore has received an impulse to move in a 
direction parallel to the axis of X, i.e. in the same direction 
in which a would have been impelled to move if the force 
had been brought to bear directly upon it. At the same 
time we have for the extremities a 1 and /3 the abscissae 
Oa + aa d x and d x respectively, and thus for the new 
position of the centre of gravity we have the abscissa 

; therefore, since the abscissa of its original 


position was it has received half of the velocity a a 

which the force applied to a tended to impart to <z, and this 
is precisely the velocity which the same force would have 
imparted to the whole mass of the system (which is a f b 
or 2 a) if that mass had been concentrated at the centre of 
gravity and the force applied to it there. 

These considerations apply to the first instant of the 
whole motion, in which (as is usually assumed) the force 
applied to <z, working instantaneously, gave ' it a certain 
velocity without any lapse of time, and in which the 
corrective reaction of the forces at work between a and b 
also took place without lapse of time. Since from this 
instant no external force any longer " operates, all the 

Chap. V.] /? TA TION PL US TRANSLA TION. 3 5 1 

motions produced will simply continue according to the lafr 
of persistence, only the internal forces that act between a 
and b have to be continually at work in order to prevent a 
and b from flying off at a tangent, and to maintain them at 
a constant distance from their centre of gravity ; they thus 
generate a rotation which is circular in relation to this point, 
and since they are continually diverting the two masses 
from their momentary direction into another without any 
breach of continuity, the rotation takes place uniformly in 
a circle and with the same constant velocity with which 
both masses are impelled in a straight line at the first 

Lastly if we move back a 1 /3, keeping it parallel with itself, 
till its centre of gravity coincides with that of a , the two 
lines will make with one another at the centre of gravity an 
angle $ equal to that which a b would make with a b at 
the point b if b were a fixed centre of rotation and the 
external force had only had to move the mass a under the 
condition that it should always be at the same distance a b 
from b. The length of the curve which a would then have 
described would have been a b . </> ; the length of the curve 
actually described by a in rotating about the centre of 

gravity which we regard as fixed is \ and this is pre- 
cisely tke velocity which the force must impart when it has 
at the same time to move the mass b in the contrary 
direction. From this we see that a momentary external 
force, whether its direction pass through the centre of 
gravity or not, always produces in the body the same sum 
of movements of translation : the rotation which is added 
in the second case is due to the internal forces which act 
between the parts of the system moved. But these forces 
are by no means inoperative even in the first case where no 
rotation occurs : but in the first case their only effect is to 
cause the several parts of the mass, which are arranged in a 
straight line at right angles to the direction of the motion 


imparted, to maintain this order during the onward move- 
ment, an effect which reveals itself in no relative movement 
of the parts about their advancing centre of gravity so long 
as we proceed upon the assumption that the body is 
absolutely rigid ; but it would at once announce itself in 
such movements if we conceived say three equal masses abc 
united to one another by pliable cords and then imagined 
an impulse to be brought to bear upon the centre of gravity 
of the whole system which lies in b. 

230. In the analysis which is required for the discovery 
of the grounds of proof we try not only to bring out the 
elements which are essential to the truth of the consequence 
to be proved, but also to eliminate those that are unessential 
for that purpose. For instance it is not uncommon in 
answering statical and mechanical questions to start from 
the supposition of a rigid line without mass. Now it may 
be granted that in the conception of a finite straight line 
the characteristic of finiteness implies the constant contact 
of each point with two neighbouring points, and the straight- 
ness implies that the line is rigid and cannot bend : only as 
a mere geometrical line it is not an object that could be set 
in motion by forces at all ; the capacity of being affected by 
forces belongs to the lineally arranged mass only, and it is 
only the forces exerted upon one another by the minute 
components of the mass that actually give to this material 
line the rigidity and unalterable length which is merely 
demanded in the geometrical conception. 

A line without mass therefore is not a happy expression, 
and does not in fact convey that which we really mean and 
upon which we build in carrying out such enquiries. A line 
must undoubtedly have mass if forces are to cause it to 
rotate about its extremity, but with a view to the laws which 
regulate the effect of these forces it is only necessary that 
the mass be the same at any cross-section of this material 
line ; any irregularity in its distribution would constitute 
a special case, in determining which we* should have to 


apply with reference to these special data the laws of thgit 
simplest case when we have the problem in its purest form ; 
on the other hand it is perfectly indifferent for these laws 
how great this mass is ; the proportions between the forces 
and the leverages necessary for equilibrium are precisely 
the same whether the lever be thick or thin, whether its 
specific gravity be greater or less. When we speak of a line 
without mass therefore we do not strictly speaking set down 
its mass as nothing but rather as a unit, and further as a 
unit to which any value great or small may be given, and 
which disappears from our further calculations just because 
as an equal factor of all the terms that stand in proportion 
to one another it does not in the least contribute to deter- 
mine or to alter the relation which subsists between them. 
This was the thought upon which the foregoing exposition 
rested. The line a b was conceived as a line of mass, and 
every one of its points as a differential of the mass : it was 
only this that made it possible to speak at all of a force W 
acting upon a <, and to set down this force W as equal to 
n co, equal to a sum of individual forces each of which was 
such as to give the velocity co to the differential of the mass. 
But we should have gained nothing by constantly taking 
count of the mass in our calculation ; only the value of o> 
would have come out differently according as the mass of 
the line or of every one of the n parts of it which we dis- 
tinguished was conceived as greater or smaller ; the relations 
between P and Q would have undergone no change so long 
as both were always related to the same mass. The division 
of the labour of proof therefore which is here introduced 
does not consist in first putting mass altogether out of sight 
and proving the law in question for the line without mass, 
and then enquiring in the second place what becomes of 
this law when mass is given to the line ; on the contrary we 
took count of this mass at the first step, but found that its 
magnitude has no influence upon the general form of the 
law: upon this ground then we may proceed in a second 



enquiry to ask how differences in the magnitude and 
distribution of the mass affect the absolute values of the 
magnitudes which are to be determined by the law. As 
soon as we take this line without mass literally and think of 
its being moved, we become involved in absurdities through 
which we can never fairly make our way, since the combina- 
tion of ideas upon which they rest is in itself an impossible 
one. What is supposed to happen when one extremity b of 
such a line receives a velocity c ? It cannot separate itself 
from the rest of the line, for then it would not be the line, 
but only the free point b that was moved : but as the line 
has received no motion how can it follow the point? It 
may perhaps be supposed that this line would rotate : then 
the point b would have to communicate its velocity to the 
other points, and that in degrees, more to the nearer and 
less to the remoter points; but we cannot see how this is 
to be measured, for all the forces are absent here which 
operating between the minute parts of a mass might cause 
the impulse received by one part to extend itself to the rest 
of the series, so that every member of it might at every 
moment receive a definite proportion of the impulse. 
Finally as there is here no reason for such an apportion- 
ment of the effect we might instead of this come to regard 
the whole line a b as a unity so closely bound together that 
every part of it, separable only to our thought or sense, 
immediately assumes the same states that are set up in 
any other part : setting aside the question whether every 
part of the line would then receive the whole velocity c or 

only - j the result would at all events be that the line a b 

remains at rest when b receives the velocity c and the other 
extremity a receives an equal velocity c. All these 
absurdities are avoided by the admission that only a line 
that has mass can be moved, not a line that has no mass. 

231. In the subsidiary processes also, the substitutions 
and transformations by which we endeavour to make the 


given circumstances accessible to our judgment, we have to 
avoid suppositions to which, however much they may help 
the imagination, no real meaning can be given. To illustrate 
this I will mention a proof which is often employed to 
demonstrate the parallelogram of forces. The body is 
supposed to move in a plane from a to c, and at the same 
time this plane is supposed to move from a to b ; and in 
this way it is fancied that the course of the body from a to 
the end of the diagonal of the parallelogram abed has been 
ascertained. This involves two assumptions which are not 
expressed but to which expression must be given ; they are 
first the assumption that the motion of the plane will not 
interfere with the motion of the point in the line a t, and 
secondly that the moving plane will carry with it the whole 
line a c together with the body. Now an empty surface in 
motion is sufficiently for removed from anything that can 
actually occur, but it is still harder to understand how a 
body can stick to it while it moves. And yet it is very 
necessary that it should so stick : for if the body be upon a 
very smooth table and we give it a push towards a r, giving 
the table at the same time a push towards a b, the body will 
not go with the table but will part company while the table 
flies away from under it. But if we supply this necessary 
condition, i.e. if we say that the body continues to move 
undisturbed towards c, while a c at the same time is com- 
pelled to move towards b and to take the body with it, the 
whole proposition becomes an empty tautology, and that 
which is assumed is precisely that which was to be proved. 
It must rank then only as one of the means which may be 
employed to give us a picture of an already demonstrated 

232. Among the numerous other proofs of the same 
proposition several proceed from a common starting-point 
which is of interest for the logician. They begin with a 
statement of the special case in which two equal forces a 
and b impel the body in two directions, and it is regarded 

A a 2 


as self-evident that the direction of the resulting motion will 
bisect the angle between these two directions. But this 
assumption includes the further assumption that if the forces 
be unequal the resultant will divide the angle into two 
unequal parts, and since it is impossible that the kind 
of this inequality should be independent of the relation 
between the magnitudes of the forces, seeing that the fact 
of the inequality depends upon it, this assumption rests 
on a more general assumption, viz. that if two conditions 
a and b tend to give each a different form to a result c, the 
recognisable influence of the two in the actual form of the 
result will be proportional to their magnitudes; if then 
a and b are equal, c will be as far removed from the result 
which would have followed from a alone as from that which 
b alone would produce. Now I cannot see why we should 
appeal to this proposition once only when we are intro- 
ducing the proof, and then conduct the proof itself by other 
complicated considerations : whatever be the forces a and b 
and the degree of their inequality, we may say universally 
that the extent to which the moved point is deflected by 
the force a from the path of the force <, and by b from the 
path of a, must vary directly as the diverting forces. In 
order to turn this logical proposition to mathematical use 
we should need first to determine how the two deflections 
are to be measured. The nature of the question does not 
invite us to apply the ordinary method and to let fall 
perpendiculars from the direction of the several paths upon 
the resultant or from the latter upon the former : all three 
paths are considered not as empty directions in space, but 
only as loci which would include the successive situations 
of the moved point. 

The following treatment is the only one suggested by this 
last remark. Let a and /3 be the two points in the paths of 
a and b respectively which the moved body would have 
reached in the same time / if it had followed the force a 
only or b only, and let /> be the point in the resultant at 


which the body arrives in the same time t under the com- 
bined influence of a and b\ then p a represents the deflection 
from the path a effected by the force , and p /3 the deflection 
from the path b by the force #, and p a : p ft = b : a. Since we 
can only estimate the magnitude of the forces a and b by 
the space which they cause to be traversed in the unit of 
time, the ratio a : b is also for the unit of time the ratio of 
the spaces traversed in the direction of a and b respectively ; 
but it must also have this meaning for any time / and for 
any part of t ; for since a and b are regarded as forces that 
operate for a moment only, the movement in the direction 
of the resultant must take place with constant velocity and 
in a straight line : the length which is traversed in the 
direction of the resultant therefore will always be propor- 
tional to the space traversed in the directions of a and b 
within an equal time /, and the lines p a and p f3 which re- 
present the deflections will form the third sides of triangles 
whose two other sides increase in the same constant ratio. 

233. But this proportion tells us nothing about the abso- 
lute magnitude of p a and p /3 ; they satisfy the proportion 
so long as they are m b and m a ; the value of this m would 
still have to be ascertained. Now there is nothing in all 
the data of the problem that can help us to determine this : 
none of them could have any influence upon it except the 
magnitude of a and <, including the ratio of a to ^, and the 
size of the included angle; but the suppositions already 
made seem to .have taken full count of the influence of these 
elements ; and it is quite impossible that anything outside 
the data of the problem can contain the grounds of some- 
thing that is to flow directly from the problem itself. In 
cases of this kind the logical course must always be to 
search for the most probable supposition that satisfies the 
requirements. The meaning to be attached to this ex- 
pression would be very hard to define in general language ; 
and my sole purpose in treating this problem is to make up 
by an illustration for the want of a precise determination of 


the general conception. The most probable supposition 
will set down that which in virtue of its nature or magnitude 
is the minimum that makes possible the relation which we 
know must subsist, and which, if it were to subsist under 
other conditions or with other subsidiary characteristics 
than those we take, would necessarily furnish special reasons 
for inferring them, which reasons are here absent. In the 
case before us the proportion p a : p /3 = b : a must always 
subsist ; therefore m cannot be nought ; but in order that 
it may subsist it is enough to set down m as equal to i ; 
and this value of ;// may be regarded as by its nature the 
minimum that satisfies the requirements; for any greater 
or smaller value, as m = 2 or /// = J, may be treated as 
m . i, i. e. as so many repetitions of the unit with the 
vanishing of which m itself vanishes and with it the whole 
relation. Unity is the only value of m which affirms that 
the required relation actually subsists in such a way as to 
enable the other special values of m to be effectively in- 
troduced as further specific characteristics, in case there be 
any reason in the nature of the content under investigation 
for preferring one of these values rather than another. Where 
as here there is no such reason we fall back upon the 
supposition that m = i, a supposition which in any case is 
necessary, and therefore is the most probable supposition ; 
for under all circumstances, even if m had some other value, 
it would hold good at the same time with that value and 
equally satisfy the required proportion. Let us then make 
the assumption and construct the figure accordingly ; i. e. 
let us from a, the extremity of the path traversed in the 
time t in the direction of #, describe a circle with radius 
equal to the path traversed in the same time towards <, and 
from /3 describe a circle with radius equal to the distance 
traversed towards a ; then these circles will cut one another 
in the diagonal of the parallelogram formed by a and b, and 
the direction and length of the resultant are both deter- 
mined at once. 


234. But even when analysis has failed to detect any 
grounds in the data of a problem for any other than this 
most probable supposition, it is seldom possible to be 
absolutely certain that such grounds are not there, and 
might not be revealed by a more careful analysis. And so 
no pains must be spared either to confirm the supposition 
adopted by subsidiary proofs upon a different line, or to 
establish it indirectly, i. e. to exclude all other suppositions 
by showing the contradictions in which they involve us. 
We will take this further step then. 

It seems self-evident that the resultant of two forces can 
never be greater than their sum ; it attains this maximum 
when they both act upon the body in the same direction, 
and when the included angle therefore is nothing. It has 
been objected to this proposition also that it is after all not 
self-evident that when a second motion b is joined to a 
motion a in the same direction b is simply added to a ; 
for it is conceivable that the nature of motion or that of 
the bodies subject to it involves conditions which might 
even in this case make the resultant greater or less than the 
sum of the two. This objection seems to me unfounded, 
especially as applied to the case before us. In the first 
place when two motions in the same direction are given at 
the same time to one body, we may continue to regard 
them as two separate motions, but it is only because we 
choose so to regard them. They were two motions outside 
the body : they may have been imparted to it for instance 
by two other different bodies. It may be also that in the 
physical act of transmission from one body to another the 
motions may lose or gain something : but we are here 
speaking not of the mode of transmission, but of the 
velocities, so far as they already have been transmitted 
to the body in question. In this body, here considered 
simply as something moveable, without regard to all its 
other peculiar properties, the two do not need to be com- 
bined into one, but they are absolutely one from the be- 


gcnning, and the resulting velocity is the sum of the two as 
surely as any velocity is what it is. But suppose the body 
already has a motion a when the second b supervenes ; this 
could not make any difference unless the body violated the 
law of persistence and altered its motion every instant : for 
if it does not alter its motion, i. e. if at the time / it is in 
precisely the same condition as at the time /, the motion 
b which supervenes later must combine with the still sub- 
sisting motion a just as it would have done at the time / 
if both had begun together. We may regard it as estab- 
lished then that the resultant R of the two forces a and b 
acting in the same direction can only be a -f b. Of course 
this does not directly help us to estimate the result of forces 
whose directions diverge and make an angle <. Meantime 
however it is at all events evident that the resultant cannot 
increase with the divergence; for then it would be least 
when the directions are the same, whereas we have just 
seen that it is greatest then, and greatest when they are 
opposite, whereas it is evident that it is least then. But it 
is equally impossible that it can be independent of the 
magnitude of the angle <f> ; and so it must necessarily 
diminish as (f> increases, and we may now say that for forces 
of any direction the resultant R is either equal to or less 
than a -f b. 

This conclusion again which is still indefinite may be 
brought within narrower limits, if we apply the important 
general principle, that objective conditions are independent 
of variations in our cognitive procedure. When various 
momentary forces to any number we please are brought to 
bear at the same time upon a moveable point, the total 
result which actually arises can only be one, and therefore 
cannot alter with the various arbitrarily chosen series in 
which we in our minds first arrange the simultaneous 
conditions by pairs, and then again combine the several 
results thus obtained. It must be the same in the end 
therefore whether we first get the resultant R out of a and b 


and then try to get a second resultant out of R and a, 9r 
whether we combine a, b, and a so that, a and a ob- 
viously cancelling each other, b is left as this second resultant. 
The conception of R therefore as the resultant of a and b 
implies that if we again take as components R and a with 
its original direction reversed and calculate their resultant 
by the^same law by which we get R from a and b, we must 
come back to b ; and so R and b combined will bring us 
back to a. And this consideration holds good universally, 
and quite independently of the still unknown law which 
regulates the dependence of the magnitude and direction 
of the resultant upon the magnitude of the component 
forces and the included angle. From this then it follows 
that each of the three forces or motions a, b, R is under the 
circumstances stated above the resultant of the other two, 
that each is therefore less than or at most equal to the sum 
of the other two ; whence it follows that the three jnay be 
combined in a triangle, which contracts itself into a straight 
line only in the limiting case where one is equal to the other 

But as thus obtained this familiar proposition only ex- 
presses a relation between the lengths of #, <, and R ; we 
must also make out the relations between the angles for 
which this relation holds between the sides. If a and b and 
the included angle $ be given, the length of R, as yet un- 
known, is completely determined ; for these given elements 
therefore there is only one possible triangle to be made out 
of a b and R. Conversely, given a triangle with a b and R 
for sides, there is but one angle <j> which the forces a and b 
can make so that R shall be the length of their resultant. 
Geometrically R in the triangle increases, if a and b are 
constant, as the opposite angle p increases ; mechanically, 
as the resultant of a and b y R diminishes as the angle <j> in- 
creases ; between the angle p in the triangle therefore and <f> 
the angle at which the forces diverge from one another there 
must subsist some definite relation which we want to as$er- 


tarn. In the triangle made up of a b and R^ R has not the 
position which it must assume when it represents the re- 
sultant ; in the latter case all three lines must start from a 
common vertex A, and it may be taken as self-evident that 
R must lie in the angle between a and b. Let us suppose 
then that a and b are two forces, as yet indefinite in magni- 
tude, put together so as to make any angle $ ; and that R 
the resultant, also as yet arbitrary in length, divides this 
angle into any two parts, C being its other extremity. Now 
as the mechanical relations of which we are here in search 
must be independent of the absolute position of the lines in 
space, we may first shift the whole system of the three lines 
a b and R so that the vertex A falls upon C, and then turn 
it, in the plane in which it lies, about C so that the forces a 
and b, which in their new position may be denoted by a 1 
and l , proceed from C in directions parallel but opposite 
to their former directions. Then evidently the resultant R 1 
of these forces a 1 and b 1 must be both in position and magni- 
tude identical with R, only opposite in direction. Thus 
then the direction of the resultant is determined ; it must 
be the diagonal of a parallelogram formed by the intersec- 
tion of the forces a and b 1 on the one side and the forces b 
and a 1 on the other, or by their meeting in a common ex- 
tremity, or by their being produced to such an extremity. 
But if the lengths of a and b are given, the length of R is 
also determined, it must be the third side of a triangle whose 
other sides are a and l , which = ^, or b and a\ which = a ; 
it is therefore the diagonal of the parallelogram formed by 
the lengths of the forces themselves. The figure then shows 
that the angle p subtended by R in either of these triangles 
is the supplement of the angle which the forces make with 
each other, i.e. that </> = TT p*. 

235. We may further confirm this conclusion indirectly 
by showing that any other supposition as to the relation 
between components and resultant is impossible. Let us 
* [See Preface.] * 


first assume that a supposition which we wish thus to test 
agrees with the foregoing so far as regards the direction of 
RI and only makes the length of R exceed or fall short of 
the diagonal Z>. Let us suppose then that the first resultant 
RI obtained from a and b is greater than the diagonal Z\ of 
the parallelogram obtained from a and b with the included 
angle & i.e. that R^~ p . D^ where/ is an improper frac- 
tion. Now if we combine this R with the force a turned 
in the opposite direction, the angle between the two being 
TT $ *, the new resultant R^ deduced from them according 
to the same supposition must be greater than the diagonal 
got from RI and a with this same angle, still greater there- 
fore than the other diagonal _Z> 2 which would be got by 
combining D^ which is less than R^ and a at the same 
angle 77 = </>*. But we know upon purely geometrical 
grounds, which are quite independent of all mechanical 
assumptions, that this diagonal D is nothing else than the 
given force b ; R< 2 then would be greater than ^, whereas we 
know for the reasons lately stated that it must be equal to b. 
If now once more we compound R^ with the given a at the 
angle <J>, the resultant R z which would be thus obtained 
must for the same reasons be equal to R l ; but by the pre- 
sent supposition it would for the angle f/> be equal to/ times 
the diagonal got from /? 2 and a at this angle ; as then R^ is 
greater than /, this diagonal also is greater than the diagonal 
D got from a and b at the same angle ; supposing it to be 
equal to q D l we get R^ = qp . D ly i. e. fi^ is q times as great 
as 7?! was. Thus the supposition that the resultant is 
greater than the diagonal leads to the absurd conclusion 
that it becomes greater and greater every time that we 
repeat this manoeuvre in its calculation. The other suppo- 
sition that it is smaller than the diagonal, i.e. that/ and q 
are vulgar fractions, would lead to an equally impossible 
diminution. In order to make this indirect proof complete 
it would be necessary to show further that the supposition 
* [TT < obviously should be it </> + the angle between A' v and .j 


of a resultant of the same length as the diagonal but making 
different angles with the given forces would involve a similar 
absurdity, viz. that its course would be more and more de- 
flected the oftener its calculation was repeated ; and lastly 
it would be indispensable to prove that there is no combi- 
nation of these suppositions in which the false consequences 
of the one would be counteracted by those of the other. 
But as the matter stands it is enough to state what the re- 
quirements of logic would be ; we may spare ourselves the 
trouble of carrying them out at length. 

236. Operations of synthesis or combination may always 
be carried out to some end, viz. to the result obtained in 
each case ; but operations of analysis on the other hand 
presuppose an end which we desire to reach, though it is 
yet uncertain whether the subject we are treating is produced 
by a combination which makes this reverse process of 
analysis possible. Even in pure mathematics therefore the 
inverse operations lead to difficulties from which the direct 
are free ; and similar doubts are suggested by the common 
practice of resolving given forces into components, though if 
the components were given no doubt would be felt about 
combining them. As any force may be split up into count- 
less pairs of components, how, it may be asked, are we 
entitled to expect that any division which we arbitrarily 
choose will have a real validity in the complex tissue of 
facts present in the problem before us ? In general terms 
this doubt is easily removed. For when we are making 
such a resolution in practice we always put one of the com- 
ponents in a direction in which some resistance or some 
counteracting force is foreseen or known to be present ; we 
only resolve therefore for convenience in formulating our 
calculation ; what we really do is to compound ; if we com- 
bine the given counterforces or resistances JFwith the given 
force F y the resultant thus got is identical with that which 
would be obtained from the uncancelled remainder of the 
onfc component of F and the whole of the other component 


which would meet with no resistance. But a real difficulty 
arises when the direction of the resistance itself is not im- 
mediately given and an attempt is made in a manner that 
seems to me hardly convincing to arrive by an application 
of the law of resolution at the principle itself which is here 
to be followed. I allude to the supposition that a plane 
resists in the direction of its normal only the imparting to it 
of a motion which makes with it any angle </>. It is quite 
easy to see that this motion may be decomposed into two, 
of which one parallel with the plane meets no resistance 
because it does not act upon the plane at all, while the 
other perpendicular to the plane is annihilated by the resist- 
ance of the plane, or at any rate is resisted by it. But how 
little right we have to carry out this decomposition here as 
one allowed by the nature of the case will appear from the 
following considerations. 

Let the moving body be a perfectly smooth ball, -and let 
it move at an angle <j> against a perfectly smooth plane E 
which offers an absolute resistance ; contact then will take 
place only in the geometrical point /, to which we must 
ascribe the same power of absolute resistance as to all the 
other points of E^ however this may be brought about. 
Now what all these other points of E have to do with the 
result which follows, it is impossible to imagine ; we think 
of them indeed when we speak of the plane E \ but as they 
are not in contact, they cannot directly contribute anything 
to the resistance, and in deducing the result we may set 
them entirely aside without altering the conditions on which 
the result is to depend. But if we do this and retain the 
point/ alone, the proposition about the resistance being at 
right angles becomes impossible, because it becomes mean- 
ingless ; for to the point p either no line is normal or any 
line drawn from it in any direction is normal. But another 
principle seems evidently to apply here : surely/, {/"it resists, 
will resist in the direction from which comes the motion to 
be resisted : there is in the first instance no conceivable 


reason for action in any other direction. If then in our 
example / were perfectly fixed, and if at the moment of 
contact the line /drawn through the point/ parallel to the 
direction of the motion did not pass through the centre of 
the ball, p would entirely annihilate the motion of that 
thread of the mass which lies in this line /; then for the 
rest of the mass of the ball, whose motion would not thus 
be annihilated, there would arise a movement of rotation, 
which would cause it to turn about the point p. The infer- 
ence that the resistance must occur in the direction of the 
motion cannot moreover be obviated by conceiving the 
moving body to be prismatic in shape, say a cube, of which 
one side remains parallel to the plane E while the direction 
of its motion makes with E the angle 4>. It is true that in that 
case two planes are brought into contact ; but even now 
every point of that part of E which is in contact will only 
be able to resist the point of the cube's side which it touches 
in accordance with the foregoing principle, i.e. in the direc- 
tion <j) ; before we could say that it would not be so we 
should have to prove that the presence of the adjacent 
points q r s of the plane E helps to determine the direction 
of the resistance offered by the point / : only this could 
render possible in fact that co-operation of the plane which 
we have hitherto spoken of, though we have not made use 
of it in deducing the result. 

And now surely it is clear that we shall never succeed in 
proving this so long as we regard E as a geometrical plane 
without physical mass and yet with power to offer resistance. 
It is not even enough to regard E as the limiting surface of 
an inert mass ; we are obliged to add a physical hypothesis 
about the forces with which the mass resists encroachment 
upon the space it occupies. We must give the plane E 
some thickness therefore \ contact will not take place at 
one point merely, but the moving body will in /act either 
penetrate to a certain depth and then be thrust back by the 
resistance of other displaced points of trie mass, or without 


coming into contact while it is still at a distance it will be 
affected by the repulsive forces of the masses united in E. 
And then we should have to prove with regard to these 
forces of all the points of the mass that in all the other 
directions they annihilate one another, but in the direction 
of the normal to the limiting surface are added to one 
another and combine to make the resistance which annihi- 
lates that component of the body's motion which lies in this 
normal but in the contrary direction. And indeed it is not 
at all surprising that we should be obliged to come back to 
an assumption of this kind : motion altogether can only 
take place in a real thing, not in a point or a line ; still less 
can we hope to calculate resistances without taking count of 
that which is alone able to resist, viz. the physical forces of 
actual bodies ; surfaces as surfaces and lines as lines always 
cut one another without any resistance at all. 

1 238. I will add one more mathematical example to 
illustrate our general directions about method. The Tay- 
lorian theorem attempts to determine the value F(x-\-ti) 
which F x^ a function of #, assumes when the variable 
quantity x increases from the limiting value which it had in 
Fx to the new value x + h. To make the statement as 
simple as possible I will subject the problem to certain 
limitations : it would take us far too long to enquire here 
whether they are superfluous or not. I conceive Fx to be 
given* in the shape of an analytical expression which indi- 
cates the mathematical operations or relations from which 
for every definite value of x flow definite values of Fx ; I 
assume that these values of Fx remain finite for every value 
of x from o to x -f ^, and that they increase continuously as 
x increases continuously between these limits. In pro- 
pounding the problem in this form, as one capable of a 
universal solution, we directly assume that the growth of 

1 I 237,which followed here, is suppressed by desire of the author as 
being altogether wrong (' wegen voliigen Irrthums durch den Verfasser 
unterdriickt'). See Editor's Preface, and Appendix. 


the function from its value Fx to its new value F( 
will follow precisely the same law which the former value 
Fx itself followed as x grew from o to its former limiting 
value X, and further that this sameness of the generating law 
will hold good for each infinitely small increment dh by 
which the function now increases precisely as for each in- 
finitely small dx by which it formerly increased. From 
this it follows that it must be possible to express either 
value of the function, and in the first instance to express 
Fx, as the sum of an infinite series, each member of which 
indicates the increase which takes place as x increases by 
the addition of each successive d x. Now if it were the 
nature of Fx that for every smallest increase of x, i. e. for 
every d x, it increased by the same constant quantity m . dx, 
its total value at the end would be the sum of an infinite 
series of similar members of the form m dx : the number of 
these members would be just as infinite as the number of 
d x into which we conceive the final value of x to be divided, 
or by the accumulation of which we conceive it to be formed ; 
the sum of the series is the integral fm dx~mx, If on the 
other hand the increase of Fx for every dx depends upon 
the value which the growing x has already attained at the 
time when this dx is added, then, if the formula we are 
seeking is to hold good for every finite x and //, the series 
we now have to take must consist of nothing but similarly 
constructed functions of x, relative successively to the con- 
tinuously increasing values of x' if we call this function/,* 
or/ 1 x, then Fx = < // 1 . d x. Now there is no reason why 
we should not repeat with regard to/ 1 * the same consider- 
ations which we have already applied toFx-, if* in/ 1 * 
now denotes a definite value out of the many values which 
x may assume,/ 1 x may also be conceived as the sum of a 
series whose infinitely numerous and similarly constructed 
members give the increments by which as each dx was 
added/ 1 x grew to its limiting value corresponding to that 
val*e of x ; and so we get/ 1 x = // 2 x. dx, and generally 


f m x = yy m +i % . d x. How to obtain from a given function 
F x these derivative functions of various grades, f l x, f 2 x> 
f m x, and how to work back from the latter to the former, 
we may assume to be well known to all who are acquainted 
with the infinitesimal calculus. 

239. These preliminary remarks really contain the solu- 
tion of the problem ; nevertheless I will proceed to trace it 
back to the following simple train of thought which may 
serve at the same time to illustrate another logical method. 

i. Evidently F(x + K) is equal to the sum of its former 
value F x and the positive or negative increment JR^ which 
F x has received in consequence of the growth of the vari- 
able x from x to x -f //. In order to determine the value 
of RI we make the simplest supposition, viz. that for each 
of these increments dh whose aggregate amounts to h, F x 
increases by the same quantity m^ d h ; then m\f dh which 
is equal to m . h is the value of R^ or is the total increase 
of F x. This m^ is not incalculable. For if, as we through- 
out assume, the increase of F x is to depend solely upon 
the nature of this function, its given value F x must have 
originated in the same manner in which its further growth 
is now to take place ; i. e. while x was passing through all 
values from o to x the function then in course of formation 
must have exhibited for each d x the same increase which 
the function thus formed now exhibits for each d h, for 
dx differs from d k in name only. Now Fx may be uni- 
versally described as the sum of a continuous series, whose 
general term is represented by/ 1 x . d x and its last term by 
the same expression if x stands for the definite limiting 
value which the variable x attains in F x. For each d x 
this series increases by/ 1 x . d x ; this quantity/ 1 x must be 
constant and be equal to m^ if the growth of Fx up to its 
given limiting value is assumed to have taken place in the 
same way S the growth from this point up to F(x -\- h). 
For every dh therefore F x increases by f 1 x . d h, and the 
sum or the integral of these elementary increments, viz. 



h ( f l x, is the required value of jR r The supposition here 
made that f l x is constant and equal to m l may not hold : 
but as the general formula must include the cases in which 
it does hold good, this second term which we have found 
may be accepted as an abiding element of it. 

2. Even if this first supposition does not hold yet 
F (x + h) is always equal to F x -f- h . f 1 x -f- R^ if we 
understand by R^ the positive or negative supplement still 
necessary for the complete measurement of the true value 
of the function. As this further addition can only be 
required because F x does not increase by the same amount 
for every dh or d x, i.e. because f 1 x is no constant 
quantity, but dependent upon the value which the variable 
x has attained at each stage, it is plain that f 1 x in the 
second term, hf l x ( R^ of our formula still denotes only 
the fixed particular value which the general function f 1 x 
now to be conceived as variable, assumes when the variable 
x assumes its limiting value x or when the variable h is 
equal to o. We cannot therefore retain this second term 
h . f 1 x unless to each of the terms/ 1 x . d h of which it is 
the sum we add the further increase exhibited by the limit- 
ing value of f 1 x contained therein for each increment d h 
of the variable h. For this increase again we make the 
simplest supposition, viz. that it is the same for each dh 
and is equal to m 2 d h. This m. 2 is also capable of determina- 
tion. For once more if our supposition is to hold good it 
must react upon Fx also \ the same law by which this 
function is now to increase must have regulated its origin ; 
the increase of f 1 x must have been the same for each d x 
and equal to m, 2 d x. Now/ 1 x is the sum of a continuous 
series whose general term is/ 2 x . d x ; this then is the very 
increment by which this series or its sum f 1 x continuously 
increases each time that x is increased by dx\ our con- 
dition is fulfilled therefore if we put down/ 2 x as constant 
and equal to m tl : then the growth of F x beyond its given 
value follows the same law which regulated its formation up 


to that point. Its total increase therefore is the sum of two 
series ; the first of these consists entirely of similar terms 
f 1 x . d h, and its sum = ^? x ; the second represented by R^ 
contains increasing terms, the first termy" 2 x . d h represents 
the first new increase which F x exhibits when the former 
limiting value x of the variable x is increased by the first 
d h, or when the variable /*, growing from o, attains its 
first value d h ; each successive (/z+i) fc h term is formed 
by adding the same increment f 1 x . d h to the value of the 
th term; h .f 2 x . d h therefore is the general term of 
this second series, and is what we must add as supplement 
to the general term of the first series. The total increase 
of F x is therefore the sum of the continuous series 

(f l x +hf*x)dh, or h.f l x-\ / 2 -#; the second term 

of this expression is the required value of R v 

3. If a given function F x were of such a nature that 
even this second supposition was not enough to exhaust its 
growth, we should still be always able to retain the terms 
of the formula already found if we added a fresh jR s to 
supplement them. And to determine this Jt 9 we should 
repeat the same process as before. We could only require 
it because f l x also is not constant, but is dependent upon 
the value which x has attained at any point and increases 
with tf. Let us assume that these increments are at least 
constant for each d h and equal to m^d h. If then we 
express f*x as the sum of a continuous series whose 
general term is / 3 x . d x, we have but to put down / 3 x as 
constant and equal to m at and we thereby make sure that 
our general condition is satisfied and that F x has grown to 
this its given limiting value in the same way as it is now to 
grow beyond it. Now R^ the third term of our formula, 
was the sum of a continuous series, whose general term is 
h .f 2 x . d k ; if then we form a second series containing the 
additions by which jR 2 is to be supplemented, h.f* x .dh 
will be the amount by which each (n + 1 ) th term of this 

B b 2 


series exceeds the n* term ; f h ./ 3 x dh therefore or 
f 2 x is the general term of this series R y We obtain 

the second and third increment of Fx therefore by sum- 
ming the continuous series whose general term is now 

* 2 

and the result is that 

! . 2. 3 

4. It would be useless to carry this process further \ it 
will readily be seen that if we constantly repeat the as- 
sumptions here made the required formula will assume the 
familiar shape of the Taylorian series, viz. 

./* +-- f*x + H * ./ 3 * ...... 

J 1.2 I . 2. 3 ' 

But this formula would be of little value if the very assump- 
tions on which it rests could not be shown to be the only 
admissible assumptions. It would be beyond all doubt 
logically correct, but only in the sense in which the 
barrenest of tautologies is correct, if it only meant that any 
quantity M might always be expressed by a series of quite 
arbitrary terms provided that we reserved the right to add a 
remaining term R intended to make good all the errors 
which we had committed by making M equal to the series. 
The formula has a serviceable meaning only when we do 
not need this compensating remainder, i.e. when we can 
prove that the value of F (x + K) can be completely ex- 
pressed either by a finite number of the developed terms, or 
by a series of such terms which though infinite yet con- 
verges so as to admit of being summed. But how do we 
learn that this is the case ? From the fact that for a given 
function F x one of its derivative functions f m x turns out 
upon actual calculation to be equal to o, that the series 


therefore breaks off before the term which contains it, *ye 
plainly can infer nothing but that there is no further increase 
of F x that can be got by the further development of the 
series we have taken ; the inference that no other increase 
can occur at all would imply that we could prove that this 
very mode of calculation must include all increase of which 
F x is by its nature capable. Now this point we think no 
longer needs special demonstration ; it is contained in the 
assumption which we made that F x does not increase 
under any other condition than that of the continuous uni- 
form increase of x y and that its mathematical structure remains 
the same for every one of the values of x which have been 
reached. If then a function grows in such a way that for 
every d h it exhibits the same constant increase, while at the 
same time every d h that thus enters into it becomes the 
starting-point of a new constant increase, we get as the 
expression of its total increase through the interval h an 
infinite series, in whose terms the one set of factors h, 

h* h 

, depend for their form simply upon this uni- 

1.2 1.2 ...m 1 l J r 

versal form of growth and are therefore similar in form for 
all functions. But in order that this series may give the 
specific growth of each particular function in distinction 
from that of any other, the other set of factors/ 1 x, f 2 x> / 3 x 
are ac^ded to these universal factors in such a way that each 
of them indicates the particular magnitude, dependent in 
each case upon the nature of the given F x y of the first, 
second, third, or m^ increase which occurs for each d h ; 
the series, as the complete expression of F (x 4- h), closes 
when one of these factors vanishes. The developed terms 
of the series above given were therefore not arbitrarily 
assumed ; what we meant to do with them was to measure 
F (x + h\ not by a standard foreign to the nature of this 
function, fcut by the standard supplied by the function itself 
and by the nature of its assumed growth ; if by this standard 
the value of F (x + h) can be expressed in a finite number 


of terms or in a number which though infinite admits of 
being summed, there can be no increase derived from other 
sources which would have to be added to this. For how- 
ever a function may grow, provided only that it is subject 
at no stage of its growth to the introduction of new con- 
ditions from without, the continued repetition of the 
assumptions above made (first of a constant increase, then 
of a constant positive or negative increase of this increase, 
then of a fresh constant positive or negative increase of this 
second increase, and so on) will enable us to exhaust the 
total value of the resulting growth just as certainly as we are 
enabled to express any curved path by properly chosen 
epicycles, or any irrational number by an infinite series of 
positive and negative powers of ten. Taken in this sense, 
as a mere definition of growth, the series remains logically 
valid even when it is rendered mathematically useless by 
divergence for a demonstrably finite increase of the function. 
If it were not so, then, even if it were possible to restore 
convergence by transforming the function without altering 
its content, the result it yielded could only be regarded as 
correct in fact, supposing it could be shown to be correct, 
it could not be regarded beforehand as obviously and ne- 
cessarily correct : such transformation only serves to bring 
within the limits of calculability what holds good as it